ABLE AND R A C T S TABLES AND TRACTS, RELATIVE TO Several Arts and Sciences By JAMES FERGUSON, F. R. S. THE SECOND EDITION, With- Additions. LONDON: Printed for W. Strahan, J. and E. Rivington, W. Johnston, T. Longman; and T. Cadell, in the Strand. MDCCLXXI. ADVERTISEMENT. Hp HE ufefulnefs of numerical Ta- •** bles, in the practical arts and fciences, is fo univerfally acknowledged, that the prefent publication fcarce needs an apology. By this means we fave a vaft deal of time and labour : witnefs the facility with which the operations of Trigonometry, and the more difficult queftions of arithmetic, are now per- formed, when compared with the ope- rofe and difcouraging methods of com- putation ufed in ancient times. No wonder therefore that the Tri- gonometrical Tables, and thofe of Lo- garithms, reduced to the moft perfect form, by the lucceffive labours of learned and ingenious men, are in every body's hands. But ftill there are many par- ticular Tables and Tracls, relative to ufeful Arts and Sciences, which lie fcat- tered in different volumes, fome in print and fome in manufcripr, to which many curious perfons cannot always have ready [ vi ] ready accefs. Such of thefe as the author judged would be m oft acceptable to the public, he hath collected into this manual, together with a few eafy rules and examples directing their ufe; To thefe he hath added feveral of his own: and, throughout the Tables, he hath taken all poflible care that the numbers fliould be correct. CON- CONTENTS.! rABLES, Precepts, and Exam- ples, for calculating the true time of New and Full Moon, in any given year and month, from the creation of the world till the 6oooth year after the end of the prefent century Page 1—5 To find the days difference between the old and new Jlile in any given year 26 To find when the Sun and Moon mufi be eclipfed 27 A Table jhewing the days of all the mean changes of the Moon, from A. D. 1 7 5 2 to A, D. 1 8 co, new Jlile, 30,31 Of the caufes and times of Eclipfes 3 2 A Table Jhewing the days of the Suns Conjunctions with the nodes of the Moons orbit, from A. D. 1752 to A, D. 1 800, new Jlile 34 A Table jhewing the Sun s place an-i de- clination 30 -4 A Tar le [ viiI ] A Table of Semi- diurnal arcs if or fhewing the times of rijing and fetting of the Sun and Moon, in all latitudes from 48 degrees to 59 45~ 53 A Table fhewing how much time is con- tained in any given number of mean lunations 5 8 A Table fhewing how many mean luna- tions are contained in any given quan- tity of time 62 An eafy method of explaining the pheno- mena of the Harvefl-moon by means of a common globe , 63 Rules for folving many aflronomical pro- blems by the logarithmic Tables of fines and tangents 67 How to draw a true meridia7i line 70 A remark concerning the placing of Sun- dials 73 How to calculate the Hour-diflances on Sun-dials 74 A Table of refraSlions of the Sun, Moon, and Jlars 79 The defcription of an inflrume7it for folv- ing many aflronomical problems ; find- ing the Hour-diflances on Sun-dials, forming forming fftherical triangles, a?id fb/v~ ing the cafes depending thereon So tT& know, by the Jlars, whether a clock goes true o?* not 105 To find the length of a pendulum that fhallmake any given number of vibra- tions in a minute 107 A Table fhewing of what length a pendu- lum mufl be, to make any given num- ber of vibrations in a minute, from 1 to i8o> in lat. 51 Q 30' 108 A Table fhewing how much a pendulum thatfwings fecond at the equator would gain every 24 hour 's in different lati* tudes; and how much the pendulum would need to be lengthened in thefe latitudes, to make it fwing feconds therein 1 1 o A defcription of three uncommon clocks in An eafy way offeprefenttng the apparent diurnal motions of the Sun and Moon in a clock 126 An eafy way of fhewing thephafes of the Moon, in a clock 127 An eafy method of fhewing the Suns place a in in the ecliptic every day of the year in a clock ; and his motion round the ecliptic in a folar year 128 How to JJjew the periodical revolutions of the Earthy and all the other planets, round the Sun, in a clock \fo as to agree nearly with the periodical revolutions of the pla?uts about the Sun in the Heavens 129 A "Table Jhewing the times of the diurnal and a?mudl revolutions of the planets, with their diflances from the Sun\ &C. 1 34 A familiar idea of the diflances of the planets from the Sun 135 Three problems Jhewing how to compute fynodical periods of any fyftem of re- volving bodies 137 How to reprefent the motions of Jupi- ter s four fatellites round Jupiter, i?i a clock ; and fhew the times of their eclipfes in Jupiter s fhadow 155 How to conflruB an Orrery for Jhewing the a?mual revolutions of Mercury, Venus, and the Earth round the Sun in their proper periodical times ; the Moons [ XI ] Moons motion round the Earthy and round her own axis, with all her dif- ferent phafes : the motions of the Sun y V e?tus y and the Earthy round their refpeclive axis ; theviciffttudes of jea- fons, the retrograde motion of the nodes of the Moons orbit ; with the times of all the new and full Moons and eclipfes 1 5 j Another Orrery "The mechanical paradox j y 2 A fhort account of the fdk mills at Derby x y ^ Rules for finding the correfponding years of the Julian period ^ with the years of the world, and years before and Jince the birth of Chrifl 175 A Table of remarkable czras and events 177 The year of our Saviour^ crucifixion afcertained y and the darknefs at the time of his crucifixion proved to be fupernatural T 8 j Tables for finding the dominical letter ana day of the month for 5600 years be- fore and after the birth ofChriJi 196 a 2 How [ *m 3 flow to divifa circles and Jiraight lines into any given number of equal parts, whether odd or even 203 How to find two vequifite points in the tube of a thermo?neter, and then to divide the fcale thereof 209 Rules for finding the areas or fuperficial contents of plane figures, and of fqlid bodies 211 Rules for finding the folid contents of bodies 222 How to gauge a common cafk or a common vat 228 A 'Table, by which the quantity and weight of water in a cylindrical pipe of any given dia??ieter, and perpe?idi- cular height of bore, may be found : and confequenily, the power that will be required to work any hydraulic engine 229 (Concerning Pumps, with a Table for Pump-makers 231 tfroy weight compared with Avoirdupoife weight 232 ^fo,ur weights, yiz. 1 pound, 3 pounds, g pounds, a?id 2 7 pounds, to weigh 4Q pounds y or any number of founds from i to 40 236 A Table of the fpecific gravities of bodies 238 A Table of the different velocities and forces of the winds 239 ^Directions for Mill-wrights 240 The Mill-wrights table i\2r The difference between the apparent level and the true 243 Of the mecha?iical power 's, and offriBion 253 A method of laying down the degrees of the Sun s declination right againjl the days of the months ; with Tables for that purpofe 262 A Table fh ewing the latitudes and longi- tudes of a great ?nany remarkable pla- ces ; and what the times are at Lo?idon when it is IVoon at thofe places 268 A Table for comparing the Englijb Avoir dupoife pound with the Foreign pound weight 274 A Table for comparing the Englijb Foot with Foreign meafures, ibid: z The [ ] The Weight and Value of Englijh Gold and Silver Coins 275 The proportion of Alloy in coinage 276 Jewifh weights reduced to Ejiglifh Troy weight ' ibid. Jewi/h-Dry meafure, Liquid- meafure, and Money y reduced to Englifh 277 A Table f jewing the Intereft of a?ty fum of moneys for any number of days y at any rate of Intereft 278 A Table fiewi?ig the Standard Weighty V alue, and comparative View of En- glijh Silver Money from the time of William the Conquer 0? *, A. D. 1066, to A. D. 1764 280 The Prices of Goods, from A. D. 1066 to A. D. 1764, reckoned according to the then current Silver Money 282 DemoivreV Tables of intereft and an- nuities 286 Tables of the Probabilities of Human Life 295 A Table fhewing how the heights of hills may be found by the barometer 302 An account of M. VilletteV burning Mirror 303 The proportional breadth of each colour in the Rain- how 304. Colours produced by the mixture of co- lour lefs fluids 306 Colours produced by the mixture of colour- ed fluids ibid. Colours changed and reflored 307 The quantity of Land and of Water on the Earth' s furf ace 308 The weight of the whole Atmofphere 309 The diameter and circumference of the vifible part of a cloudy Jky 310 The velocity of Light ibid. The velocity of Sound 311 The caufe of the ebbing and flowing of the fea at the fame time on oppojite fides of the globe , accounted for ibid. Surprijing properties of numbers^ placed in fquares and circles 317 A magic fquare of fquares 318 A magic circle of circles 320 Table [ 1 ] Table I. The mean time of New Moon in jfa nuary^ according to the Old Stile. New Moon. D.H.M. 1700 8 14 45 01 27 12 17 02 16 21 6. 03 6 5 55 04 24 3 27 °5 13 12 16 . 06 2 21 4 07 TA 21 I 8 37 OS 10 3 25 O9 29 0 58 I7lO 18 l 9 47 I ! 7 18 35 12 2 5 16 3 13 J *5 0 56 H 4 9 45 fs z 3 7 17 16 11 16 6 17 1 0 54 18 19 22 27 19 9 7 16 1720 z 7 4 4$ 21 16 x 3 37 22 5 22 2 S - 23 24 19 rg 24 13 4 4 6 2 5 2 '3 35 26 21 I i 8 27 10 19 56 28 28 I? 29 29 18 2 17 1730 7 U 6 3 1 26 8 38 32 H 17 27 Sun's Anomaly 6 21 14 7 10 7 6 29 22 6 18 38 7 7 1 6 26 16 6 1 c 32 7 3 35 6 23 11 7 1* 33 7 0 48 6 20 3 7 8 26 6 27 42 6 16 57 7 5 20 6 2 4 35 6 13 51 7 2 13 6 21 29 7 9 5? 6 29 7 6 18 23 7 6 45 6 26 1 6 *5 17 7 3 39 6 22 55 7 n 17 7 0 33 6 19 49 7 8 11 6 27 27 13 Moon's Anomaly. S. 0 ' 0 0 55 1 1 6 32 9 16 20 7 23 8 7 1 45 5 11 33 3 21 21 2 26 58 1 6 46 0 12 23 10 22 12 9 2 0 8 7 37 6 17 2? 4 27 13 4 2 50 2 12 38 0 22 26 11 28 3 10 9 13 28 7 23 16 6 3 4 5 8 41 3 18 29 1 28 17 1 3 54 1 1 13 42 10 19 19 8 29 7 7 8 55 6 H 33 4 24 20 Sun from Nod 9. S. 4 13 9 5 21 52 5 2 9 55 6 7 57 7 16 41 8 24 43 8 2 46 9 11 29 9 19 32 10 28 15 1 1 6 18 1 1 14 20 0 23 3 1 1 6 1 9 9 2 17 52 2 2 5 54 3 3 37 4 12 40 4 20 43 5 29 26 6 7 29 6 *5 3 1 7 24 H 8 2 17 8 10 20 9 19 3 9 27 $ 1 1 s 4sr 1 1 *3 5? 1 1 31 54 1 0 37 1 8 40 [ 2 ] Table I. continued* Old Stile. New Sun's Moon's Sun from Juli i ea Moon. Anomaly. Anomaly. Node. ? 3 D. H. M. S. 0 , S. 0 S. 0 1733 A. f 2 16 6 16 4"? % j 4 Q y 1 16 43 TJ 34 22 2? j 48 7 j c J 2 Q 46 2 2C j 26 3 C I 2 8 57 6 14 21 0 I Q 34 3 3 28 2 6 J O / 2 ^ 6 J 36 J 10 20 22 '311 3 I 37 J 1 10 14 T <;8 j 7 I ?8 9- 10 A 1 3 y : 4 20 14. 38 8 23 47 6 21 14 8 14 T 47 4 28 17 3Q 2 7 * / 2 I I 0 7 9 36 7 20 24 6 7 O I 74.0 16 6 8 6 28 C2 J 6 O I 2 6 i.c A I c J 14 56 6 18 Q y 4. T 10 O 6 2 3 r ? 42 r 24. 1 2 2Q 7 6 30 J T C * 5 37 8 1 48 43 I 2 2 1 1.8 6 2C j 46 T I 2? 2.C 4 Q C I J 41' 2 6 6 6 2 O i r I 3 8 17 C4 2 I ■3 J 1 •2 j 24, T I I I O CO 38 9 26 37 10 I 2 27 6 22 50 Q 20 10 4 AO 4.7 T' 29 IO IO 7 1 1 2 8 26 16 11 13 23 4-8 l 7 18 48 7 0 1 7 7 6 4 11 21 2C 49 7 3 37 6 1 9 33 5 15 11 29 28 1750 26 1 10 7 7 55 4 21 2C 1 8 I I 5 1 15 9 58 6 27 1 i 3 1 1/ 1 16 M 5 2 3 18 47 6 16 27 1 1 1 5 1 24 17 S3 22 16 19 7 4 49 0 16 4 2 3 3 0 54 1 2 1 8 6 24 4 10 26 3^ 3 11 2 55 1 9 56 6 13 21 9 6 18 3 '9 5 56 *9 7 29 7 1 43 8 11 55 4 2 7 48 57 8 16 17 6 20 59 6 21 43 5 5 51 58 27 13 5° 7 9 21 5 27 z r 6 14 34 59 16 22 39 6 28 36 4 7 8 6 22 37 1760 5 7 27 6 17 5 2 2 16 5 f 7 0 39 61 24 5 0 7 6 14 1 22 33 8 9 22 62 13 13 48 6 3° 0 2 21 8 17 25 63 2 22 36 6 46 10 12 9 8 25 28 64 20 20 9 7 3 8 9 «7 46 10 4 1 1 65 10 4 58 6 22 24 7 27 35 10 12 H 66 29 2 3P 7 10 46 7 3 12 I 10 20 57 [ 3 ] Table I. concluded. Old Stile. New Sun's Moon's Sun from Julian Years, Moon. Anomaly. Anomaly. Node. H. M. S. c 0 c 0 0 ✓ .. \ -. 1767 18 1 1 X 9 7 O 2 5 l 3 0 u 28 59 68 6 20 8 0 *9 1 b 3 2 2 4 b 072 69 2 5 40 7 7 4 6 • 2 2o 2 5 1 *5 45 1770 15 2 29 O 26 56 1 0 0 •3 1 23 48 71 4 1 1 1 7 O 1 6 1 1 1 1 I 0 2 1 51 72 22 0 0 5° 38 7 4 33 1 0 2 3 3 s 3 *° 34 73 1 1 17 0 2 3 49 9 3 20 3 »8 36 74 1 2 27. 6 13 5 7 *3 14 3 26 39 75 20 0 O 7 1 27 /: 0 5 1 5 5 22 76 8 0 48 6 20 43 4 2 0 39 5 »3 25 77 27 O 21 7 9 5 4 4 10 6 22 8 78 16 15 9 6 28 21 2 H 4 £ 8 U 79 5 23 58 6 17 37 0 2 3 5 2 7 8 13 1780 2 3 2 1 3° 7 5 59 1 1 29 29 8 16 56 81 13 /T O *9 6 25 15 10 9 *7 8 24 59 82 2 15 7 6 H 3° Q O 19 5 9 3 2 0 0 2 1 1 2 4.0 7 2 5 2 7 z 4 42 10 1 1 45 84 9 21 29 6 22 Q f. 0 4 3 1 10 x 9 48 85 28 19 I 7 10 J c J 10 8 11 28 31 •86 18 3 50 6 29 46 3 19 5 6 0 6 33 •87 7 12 38 6 19 2 1 29 44 0 14 36 » 58 »5 10 II 7 7 24 1 5 21 8g 14 18 59 6 26 40 11 *5 9 2 1 22 1790 4 3 48 6 *5 5? 9 24 57 2 9 2 5 91 23 1 21. 7 4 18 9 0 34 3 18 8 92 1 1 10 9 6 2 3 33 7 10 22 3 26 10 93 0 18 58 6 1 2 49 5 20 10 4 4 *3 94 *9 16 3 C 7 1 11 4 2 5 47 5 12 5 6 95 9 1 *9 6 20 z 7 3 5 3 5 5 20 59 96 26 22 51 7 8 49 2 1 1 12 6 29 42 97 16 7 40 6 28 5 0 21 0 7 7 45 98 5 16 29 6 '7 21 1 1 0 48 7 x 5 47 99 24 H 1 7 5 43 10 6 25 8 24 30 1800 12 22 5° 6 24 59 8 16 1 9 2 33 B 2 [4] Table II. Mean New Moon in January, New Stile. 2 C« 1752 53 54 55 5 57 5* 59 17 'C 6 62 63 6^1 6 S 66 67 68 % i77o 71 72 73 74 75 70 79 1780 81 82 83 84 85 86 New Moon. D. H. M H 18 47 4 3 55 z$ I 8 (2 956 30 7 29 19 l6 17 9 1 6 27 22 39 1 6 72 ; 16 16 24 13 48 '3 22 37 7 4 1 5 I i 20 4 5 6 2 49 II 17 2 10 2> 9 l -7 7 2') »5 3 20 22 I7 19 8 2 27 II 15 8 48 17 37 27 15 9 16 23 58 5 8 4* 24 6 19 '3 15 7 z 23 56 20 21 29 10 6 17 29 3 <° Sun's Anomaly. 16 27 S 4* 24 5 13 21 1 43; 20 59 10 14 28 36 17 52 7 8 25 30 H 46 4 2 22 24 1 1 40 o 2 18 l 9 8 ' 3'3 26 56 16 11 5 2 7 6 23 49 6 13 5 2 21 20 43 9 59 28 21 17 37 6 52 25 *5 14 30 3 46 22 8 1 1 24 29 46 Moon's Anomaly S. in 5 11 20 53 10 26 30 6 18 11 55 zi 43 1 31 7 8 16 56 26 44 2 2 1 10 12 9 8 21 57 7 27 35 6 7 23 5 13 0 3 22 48 2 2 36 1 813 11 18 1 9 9 7 5 4 3 2 o 1 r 10 8 6 6 4 3 27 49 3 26 J 3 '4 23 2 28 39 8 27 14 4 23 52 28 S3 4 31 14 19 19 c;6 Sun from Node. - S. 0 1 24 17 2 219 3 11 2 3 19 5 4 2 7 48 5 5 ?i 5 13 54 6 22. 37 0 39 8 42 7 7 8 17 2 5 8 25 28 9 3 3 1 10 12 14 10 20 16 28 59 17 2 1 23 48 i 5i 9 53 18 36 39 42 26 4 4 5 13 2 5 5 21 28 7 on 7 8 13 7 16 16 8 24 59 9 3 2 9 I* 5 10 19 48 10 27 50 o 6 33 E s ) Table 11. concluded* New Stile, i 7 8 7 88 179° 91 92 93 94 95 96 97 98 99 New Moon. D. H. M. 18 12 38 6 21 26 2 5 18 59 15 3 48 4 12 37 22 10 9 1 1 18 58 1 3 46 20 1 19 8 10 7 27 7 40 16 16 29 6 1 '7 24 22 5 C Sun's Moon 's Sun from Anomaly. Anomaly. Node. S. 0 S. 0 ' s 0 OI9 2 1 29 44 0 T 36 6 8 18 0 9 32 0 22 39 6 26 40 11 15 9 2 I 22 6 1 5 56 9 24 57 2 9 2? f, " T 1 (J S * 1 8 4 45 2 *7 27 6 23 33 7 10 22 3 26 10 6 12 49, 5 20 10 4 4 13 625 3 29 58 4 12 16 6 20 27 3 5 35 5 20 59 6 9 43 1 >S 23 5 29 2 6 28 5 0 21 0 7 7 45 6 17 21 11 0 48 7 15 47 6 6 37 9 10 37 7 2 3 5° 6 24 59 8 16 9 2 33 Table III. Containing 1 3^ mean Lunations, 1 2 3 4 5 6 7 8 9 10 II 1.2 13 New Moon. D. H. M. 29 12 44 59 I 28 88 14 12 2 56 15 40 4 24 17 8 1 4 2 18 36 7 20 20 5 8 49 21 33 18 22 118 ,H7 177 206 236 •265 295 3 2 4 354 383 17 Sun's Anomaly. S. 0 29 6 1 28 13 2 27 19 3 26 25 4 2 5 3 2 5 2 4 3 8 6 23 44 7 22 51 8 21 57 9 21 3 10 20 10 11 19 16 O l8 22 o 14 33 Moon's Sun from Anomaly. Node. S. ° ' S. 0 f 0 25 49 1 0 40 1 21 38 2 1 20 2 17 27 3 2 1 3 13 16 4 2 41 4 9 5 5 3 2 ' 5 4 54 6 4 1 6 0 43 7 4 42 6 26 32 8 5 22 7 22 21 9 6 2 8 18 10 10 6 42 9 *3 59 11 7 2 3 10 9 48 0 8 3 n 5 37 | 1 8 43 6 12 q4 • 0 ic 20 Table IV. Supplemental to Table L for finding the mean time of New Moon in Ja- nuary, for booayears before or after any given year of the 18 th Century according to the Old Stile, o New S un' Moon's Sun from Juii, ntui Moon. Anomaly. Anomaly. Node. "ies. D. H. M. S. Q S. 0 S. 0 ' IOO 4 8 16 5 0 3 *5 8 *9 4 19 24 200 8 10 0 6 29 5 0 37 9 8 48 300 «3 0 0 9 44 1 >5 5 6 1 28 12 400 17 8 *9 0 1 2 59 10 1 15 6 17 36 500 21 ID 24 0 16 *3 6 16 33 1 1 7 0 600 26 O 29 0 19 28 3 1 5 2 3 26 24 700 0 19 5° 1 1 23 36 10 21 22 7 15 8 800 5 3 55 1 1 26 51 7 6 4b 0 4 31 900 9 1 1 59 0 0 16 3 21 59 4 2 3 55 iooo- 1 3 20 4 0 3 20 0 7 18 9 I IOO 18 4 9 0 t> 3 5 3 22 37 2 2 43 I200 22 12 H 0 9 49 5 7 55 6 22 7 1300 26 20 *9 0 13 4 1 23 H 1 1 11 31 1 400 1 15 40 1 1 17 2 9 1 2 44 3 0 15 1 500 5 53 44 1 1 20 2 7 5 28 2 7 '9 39 1600 10 7 49 1 1 23 42 2 *3 21 0 9 3 1700 H '5 54 1 1 26 56 10 28 40 4 28 27 1800 18 23 59 0 0 1 1 7 *3 58 9 »7 5i 1900 2 3 8 4 0 3 26 3 29 »7 2 I *5 2 CCO 27 16 9 0 6 40 0 H 36 6 26 39 2 I CO 2 1 1 z 9 1 1 10 49 0 u 4 5 10 15 2 3 2200 6 '9 34 1 1 14 3 4 *9 24 3 4 47 2300 1 1 3 39 1 1 17 iS 1 4 43 7 24 u 24OO 15 1 1 44 1 1 20 33 9 20 1 0 13 34. 25OO 19 19 49 1 1 2 3 47 6 5 20 5 2 58 260O 2 4 52 1 1 27 2 20 39 9 22 22 270O 28 1 1 58 0 & 17 1 1 5 57 2 1 1 46 280O 3 7 *9 1 1 4 2 6 2 5 27 6 0 30 29OO 7 1 5 24 1 1 7 40 3 10 46 10 19 54 3OCO 1 1 21 29 1 1 10 54 1 1 26 4 3 0 18 C 7 ] Table IV. concluded. New Sun'. Moon's Sun from « , j 1 ~ Moon. Anoma Anomaly. Node. D. H. M. S, 0 '* S. 0 ' S. 0 3 ico 16 7 34 1 1 9 8 11 23 7 28 42 2 200 J 20 15 5b 1 1 17 24 4 26 42 0 18 16 24 2 3 43 1 1 20 3 8 1 12 1 5 7 30 3400 29 7 48 1 1 2 3 53 9 27 19 9 26 54 3 coo 4 3 9 10 28 1 5 16 49 1 15 38 3600 8 11 14 1 1 1 16 2 2 8 6 5 2 3700 1 2 19 19 1 1 4 30 10 17 26 10 24 26 3800 "7 3 2 3 1 1 7 45 7 2 45 3 13 50 39CO 2 1 11 28 1 1 1 1 0 3 18 4 8 3 H 4000 2 5 19 33 1 1 14 '4 0 3 22 0 22 37 4IOG 0 H 54 10 18 23 7 22 52 4 1 1 21 4200 4 22 59 10 21 37 4 8 11 9 0 45 4300 9 7 4 10 24 s"2 0 23 29 1 1 20 9 A 4.00 13 15 8 10 28 7 6 8 48 6 9 33 4500 17 23 '3 1 1 1 2 1 5 24 7 10 28 57 460O 1 22 7 18 1 1 4 2 9 25 3 18 21 4700 26 15 2 3 1 1 7 ? 1 10 24 44 8 7 45 48OO 1 10 44 10 1 1 s"Q 6 14 14 1 1 26 29 49OO 5 18 48 10 15 2 29 32 4 *5 53 5000 10 2 59 10 18 28 1 1 H 5 1 9 5 17 5IOO U 10 58 IO 2r 43 8 O IO 1 24 4i 5200 18 19 3 10 24 58 4 15 29 6 H 5 5300 2 3 3 § 10 28 12 1 0 47 1 1 3 29 5400 27 11 13 1 1 I 2 7 9 16 6 3 22 53 5,-00 560O 2 6 33 H 38 10 5 35 3 *5 3 6 7 11 36 6 10 8 50 1 20 54 0 1 0 5700 10 22 43 10 12 5 10 16 13 4 20 24 58OO 15 6 48 10 *5 l 9 6 21 32 9 9 48 5900 j 19 »4 53 10 18 34 3 6 50 1 29 12 6pooj 23 22 58 10 21 48 1 1 22 9 6 18 36. The centurial differences in this Table are equal, but in Lunations themfelves they are not. — The follow- ing Table fhews the centurial va- riations. C 8 3 Ta ble V, V aviations in the mean times of New and Full Moons for 30 centuries, both before and after the 18 th century* New uUU iium 0 B <— Moon. A nnir* alu Node* s ~- Add. 0 u 0 1 rcici . • H. M. S. 0 ' " 1 CO O O zC 0 024 Q O 0 c zco 0 i 40 O 032 •300 u 3 4|> u 3 3 U O I I 2 400 0 6 ^0 O Z 0 IJOO O E O 2 ^ O I O O O 3 20 6co O I £ O O 14 24 u 4 4 700 O 20 2 CJ 0 632 800 u ^5 30 0 8 32 900 0 33 45 0 3 : 2 0 10 48 1 000 O ^. I 0 40 0 O 13 20 I ICO O *JO 2 ij 048 24 OIO 0 I 200 IOO 0 5/ 3° O I9 12 I I O 2 C[ I73O O 2 2 3 2 I 4GO I 21 ^.O T 1 ft -7 1 10 ^ 4- O ZO 0 I5OO 1 33 45 1 30 0 O 30 O i6co 1 46 40 1 42 24 0 34 8 1700 2 0 25 1 5 5 3 6 0 38 32 1800 2 15 0 2 9 36 0 43 12 1900 2 30 25 2 24 24 0 48 8 2000 2 46 40 2 40 0 0 53 20 2100 3 3 45 2 56 24 0 58 48 2200 3 zi 4c 3 '3 36 1 4 3? 2300 3 40 25 3 3 1 3 6 1 10 32 2400 400 3 5° 2 4 1 16 48 2500 4 20 25 4 10 0 1 23 12 2600 4 41 40 4 3° 2 4 1 30 8 2700 5 3 45 4 51 36 I 37 12 2800 5 26 40 5 n 3 6 2 44 32 29c© 5 5° 2 5 5 36 24 1 52 8 3000 0 ii; 0 60c 200 3 ^3" £-rt g H 1 — ' 3^ ^r- m ti - 3 c £ n> 3 Si Si * cr 3" rj rt cr£ 5t P p » v 1 u n> p p cr o n> rt cu 1/5 3^ >-• H n so rt h3 Cr cr > O 3 3?° § - & b p- o P f9 Cu o o o ^ E-3 rt ^ 3-" rt rt ^OP 3 0 < n> ^ m » P 3 3 [ 9 ] 1 'able VI. The days a common year y reckoned from the beginning of fanuary^ and ferving (with the foregoing Tables) to find the days of New a?id Full Moons in all the other months. u » t> % c "it > CO 0 • 3 " 6> T3 "i n e °? *X3 r s> 0 0 I 1 3 2 C OO 9 1 121 152 182 213 244 274 3°5 335 2 2 33 OI 92 122 i53 183 214 245 275 306 336 3 3 34 /T _ 02 93 123 *54 184 215 246 276 3°7 337 4 4 35 6 3 94 I24 155 185 216 247 277 308 338 5 5 3 6 64 95 125 156 186 217 248 278 3°9 339 6 0 37 6 5 96 126 x 57 187 218 249 279 310 340 7 7 3 8 66 97 127 158 188 219 250 280 3 11 34i 8 0 0 39 67 98 128 '59 189 220 251 281 312 342 9 9 40 68 69 99 I29 160 190 221 252 282 313 343 IO 10 4 1 100 I30 161 191 222 253 283 3H 344 1 1 1 1 42 70 101 131 162 192 223 254 284 3 I 5 345 12 12 43 71 102 132 163 193 224 2 55 285 316 34 6 *3 44 72 103 x 33 I&4 194 22 5 2oO 317 347 14 *4 45 73 104 134 I65 ■95 226 2 57 287 3»8 348 15 *5 46 74 105 135 l66 196 227 258 288 3i9 349 l6 16 47 75 106 136 I67 197 228 259 289 320 35° 17 *7 48 76 107 i37 l68 198 229 260 290 321 35 1 18 18 49 77 108 138 I69 199 230 261 29I 322 352 19 l 9 5° 78 109 139 170 200 231 262 292 3 2 3 353 20 20 5i 79 1 10 140 171 201 232 263 2 93 324 354 21 21 52 80 1 1 1 141 172 202 233 264 294 3 2 5 355 22 22 53 81 112 142 l 73 203 234 265 295 3 26 3 5 6 2 3 23 54 82 "3 M3 '74 204 23c; 266 296 327 357 24 24 55 83 114 144 175 2C 5 256 267 297 32s 3S8 2 5 25 56 84 115 *45 i 7 6 200 237 268 298 329 359 26 26 57 85 116 146 177 207 238 269 2 99 33° 3 to 2 7 27 58 86 117 H7 .78 208 239 270 300 33 1 361 28 28 59 87 118 148 179 209 240 271 301 332 362 29 29 88 119 149 180 210 24 1 272 302 333 363 30 3° 89 120 150 181 211 242 273 3°3 334 364 31 3 1 90 '5i ] 212 304 16$ c [ IO ] TableVII. Fir /I Equation from mean to true Syzygy. / rgument Sun's mean Anoma!\ Subtract. D 0 1 2 3 _ 4 5 ro crq S igns. Sign. Signs. Signs. Signs. Signs. ro to to - to f I i.M. H.M. H.M. H.M. H.M. H, M. - ? O 0 0 2 3 _____ 3 35 _ 4 11 3 4° 2 8 30 I 0 4. 2 7 3 37 4. 1 1 • 3 37 2 4 29 2 0 9 2 I I 3 39 4 11 3 35 2 0 28 3 0 13 2 14 3 4 1 4 11 3 33 1 56 27 4 0 17 2 18 3 43 4 11 3 3° 1 52 26 5 0 21 2 21 3 45 4 10 3 28 1 48 2 S 6 0 26 2 25 3 47 4 10 3 26 1 41 24 7 0 29 2 28 3 49 4 10 3 2 3 I 4O 2 3 8 0 34 2 32 3 5 1 4 9 3 20 I 36 22 9 0 38 2 35 4 9 3 18 I 32 21 IO 0 43 2 39 3 54 4 8 3 *5 I 28 20 1 1 O 4.7 2 4.2 3 c6 4 7 3 12 I 23 19 12 O C I j 2 AC ~ j 3 <\7 4 6 3 9 I IQ 18 1 3 J 0 cc J J 2 48 3 s8 4 6 3 6 17 14 0 59 2 52 4 0 4 5 3 3 1 11 1 6 15 1 4 2 5C 4 1 4 4 3 0 1 6 15 l6 1 8 2 58 4 2 4 3 2 57 1 2 H 17 1 1 2 3 1 4 3 4 J 2 54 0 58 •3 18 1 16 3 4 4- 4 4 0 2 CI j 0 ci 12 19 1 20 3 7 4 5 3 58 2 47 0 49 1 1 20 1 24 3 10 4 6 3 57 2 44 0 44 10 21 1 28 3 12 4 7 3 5 6 2 41 0 40 9 22 1 3 2 3 15 4 8 3 54 2 37 0 36 8 2 3 1 36 3 18 4 8 3 53 2 34 0 31 7 24 1 40 3 20 4 9 3 5 1 2 30 0 27 6 2 5 1 44 3 23 4 9 3 49 2 26 0 22 5 26 1 48 3 26 4 10 3 48 2 23 0 18 4 27 1 52 3 28 4 10 3 4 6 2 19 O 15 3 28 i 56 3 3° 4 >• 3 44 2 15 0 8, 2 29 1 59 3 33 4 n 3 4 2 2 1 2 0 4 1 30 2 3 3 35 4 i 1 3 4° 2 8 0 c 0 ro 1 1 10 . 9 8 . 7 6 O Cro Storis Signs. Signs. Signs. Signs. Add. [ » ] 1 able VIII. Equation of the Moons mean Anomaly. Argument, bun's mean Anomaly. Subtract. m s> ro 0 2 , 3 4 . 5 r— 1 Signs. C 1 rrn Signs. Signs* Signs. Signs. a 1 n • - — — — f& H. M. H.M. H.M. H. M. H. M. H.M. * * ■~ — ■ — o 0 0 5 l 3 8 47 9 47 — 8 9 4 35 30 I 0 11 5 22 8 52 9 4 6 8 3 4 26 29 2 0 21 5 3i 8 56 9 45 7 57 4 17 28 3 0 33 5 4° 9 0 9 44 7 54 4 9 2 7 4 0 44 5 49 9 5 9 43 7 46 4 0 26 5 0 55 \ S l 9 8 9 4 2 7 4° 3 5i 2 5 6 I 6 6 6 9 12 9 4° 7 34 3 43 24 7 1 17 6 14 9 16 9 3 8 7 2 7 3 34 2 3 8 1 28 6 23 9 l 9 9 3 6 7 21 3 2 5 22 9 1 39 6 31 9 22 9 3 + 7 »4 3 i 6 21 IO 1 5c 6 39 9 2 5 9 3 2 7 8 3 7 20 1 1 2 c 6 47 9 28 9 3° 7 » 2 58 »9 12 2 11 6 55 9 31 9 2 7 6 54 2 49 18 "3 2 22 7 2 9 33 9 2 4 6 47 2 40 17 >4 2 33 7 10 9 35 9 21 6 40 2 30 16 »5 '2 43 7 J 7 9 37 9 18 6 33 2 21 •5 16 2 54 7 2 4 9 39 9 *4 6 26 2 12? *4 1 / 3 4 n 2 1 9 1 1 6 18 1 2. 18 3 H 7 38 9 4 2 9 7 6 11 » 54 12 *9 3 2 ' 7 45 9 44 9 3 6 3 1 44 1 1 2C 3 35 7 5 1 9 45 8 59 5 5 6 1 35 IO 21 3 45 7 5 8 9 47 8 55 5 48 1 26 9. 22 3 55 8 4 9 47 8 50 5 40 1 16 8 2 3 4 5 8 10 9 47 8 46 5 3 2 1 7 7 24 4 *'5 8 16 9 48 8 41 5 2 4 0 57 6 4 2 5 8 21 9 48 8 36 5 16 0 48 5 26 4 35 8 27 9 48 8 3^ 5 8 0 38 4 2 7 4 45 8 32 9 48 8 26 5 0 0 29 3 28 4 54 8 37 9 48 8 20 4 5 1 0 19 2 29 5 4 8 42 9 47 8 15 4 43 0 10 1 20 5 J 3 8 47 9 47 8 9 4 35 0 0 0 '© ft 1 1 10 . 9 8 Signs. .7 6 O °P Signs. Signs. .Signs. Signs. Signs. crq Subtraft. C ii ] I. To calculate the true time of New or Full Moon> in any given year and month of the i$th century. From Table I. (page i, 2, 3) write out the mean time of New Moon in January, Old Stile, for the given year, with the mean Anomalies of the Sun and Moon, and the Sun's mean diftance from the afcending Node of the Moon's orbit. If you want the time of Full Moon, add the half lunation, with its Anomalies, Sec. at the foot of Table III. (page 5) to the forefaid numbers, if the New Moon taken, out falls before the 15th of January ; but if it falls after, fubtracl the half lunation, &c. from the faid numbers ; and write down the refpedive fums or remainders.* If you want to calculate for the New Stile, in any given year from A. D. 1752 to 1800, take out the Mean New Moon with its Anomalies, &c. from Table II. (page 4, 5) In thefe additions, or fubtractions, remember that 60 minutes make a degree, C '4 ] degree, 30 degrees make a lign, and 12 figns make a circle* And that, when the number of figns you fubtract from is lefs than the number of figns to be fub- tra&ed, add 12 figns to the leffer num- ber ; and then you will have a remain- der to fet down. A Jign is marked thus f , a degree thus °, and a minute thus \ When the required New or Full Moon is in any given month after Ja- nuary, add as many lunations from Ta- ble III. with their Anomalies, &c. to the numbers taken out for January, as the given month is after January ; fetting them in order below the Ja- nuary numbers : and thefe added to- gether will give the Mean time of New or Full Moon, with the Anomalies thereto belonging, for the month de^ fired. With the number of days added to- gether, enter Table VJ. (page 9) un- der the given month ; and againft that number you have the day of New or Full Moon in the left hand column (under [ is ] (under Days) which you are to fet be- fore the hours and minutes already found. But, as will fometimes happen, if the faid number of days falls fliort of any in the column under the given month, add one lunation and its Ano- malies to the forefaid fums ; and with this new number of days enter Table VI. under the given month, where you are fure to find it the fecond time, if the firft falls fhort. Then, with the figns and degrees of the Sun's mean Anomaly, enter Ta- ble VII. (pag. 10) and therewith take out the firft Equation from mean to true Syzygy^ making proportions in the Table for the minutes of Anomaly above whole degrees, becaufe the Ta- bles give the Equations only to whole degrees. Subtract this Equation from the mean time of New or Full Moon, if the figns are at the head of the Ta- ble, in which cafe the degrees are in the left hand column and reckoned downward: but if the figns of Ano- maly are at the foot of the Table, in which C is ] which cafe the degrees thereof are irt the right hand column, and reckoned upward, add the Equation to the above found time of New or Full Moon. With the figns and degrees of the Sun's mean Anomaly enter Table VIII. and therewith take out the Equation of the Moons mean Anomaly \ and ap- ply that Equation to the Moon's mean Anomaly, fubtrading it therefrom if the figns are at the head of the Table, and their degrees at the left hand ; but adding it to the mean Anomaly of the Moon, if the figns of the Sun's Ano- maly be at the foot of the Table, and their degrees at the right hand ; and you will have the Moon's equated Anomaly ; with which enter Table IX. and take out the Equation anfwer- ing thereto, adding it to the former equated time, if the figns are at the head of the Table, but fubtra&ing it therefrom, if they are at the foot ; and the refult will give the true time of the required New' or Full Moon, near enough for any common Almanack. n The [ '7 § The Tables begin the day at Noon, and reckon the hours and minutes thence forward to the noon of the following day. They give the right time in all the months of common years, and in all the months after Fe- bruary in Leap years. But in Janu- ary and February, in Leap years, a day muft be added to the time given by tke Tables. Example I. For the true time of Full Moon in March 1764, New Stile, By the Precepts. New Moon. Sun's Anom. Moon's Anom. Sun from Node. D. H. M. S. ° ' S. ° ' S. 0 ' 1 To Jan. 1764 I add * Lun. 2 Lunations, 2 7 25 14 18 22 59 * 28 .642 0 14 33 1 28 13 8 21 57 6 12 54 1 21 38 9 3 3i Q 15 20 2 I 20 Full ]) Mar. FiiftEqu. *7 3 *5 +4 6 8 16 48 Ar. 1 Eq. 4 26 29 + 1 33 II 20 II Second Equ. 17 7 21 + 4 5° 4 28 2 Ar, 2 Eq. ' Trie time 17 12 II D By By this fhort procefs it appears, that the true time of the required Full Moon was the 17 th of March, at 11 minutes paft 1 2 o'clock at night. A few more examples will make the whole matter plain. Example II. For the true time of New Moon in April T 1764, New Stile. By the Precepts. New ■(Moon. - bun's Anom- Moon's Anom. Sun from Node. D. H. M. s. ° r S. 0 ' S. ° ' Jan. 1764 -f 3 Lun. 2 7 88 14 12 642 2- 27 19 8 21 57 2. 17 27 9 3 3* 321 March Firft Equ. 31 21 37 + 4 11 9 I 21 Ar. 1 Eq. 11 924 + 1 35 0 5 32; Sec. Equ. 32 1 4 3 —3 2 5 11 10 59 Ar. 2-Eq. True time 3 1 22 2% This fhews the true time to be at 22 hours, 23 minutes, after the noon of the 31ft of March ; which is the ift of April at 23 minutes paft X in the morning. [ ] 31. calculate the true time of New or Full Moori) in any given year and months of any century y between the Chriftian JEra and the 1 8 th Centu- ry. Old Stile, In Table I. find a year in the 18th Century, of the fame number with that in the Century propofed, and take out the numbers belonging thereto as in the preceding Examples. Then, from Table IV. take out the numbers an- swering to the number of Centuries before the 18th, fubtra&ing them from •thofe of the 1 8th, and fetting down the remainder. To this remainder join the numbers for as many Centuries, from Table V. Subtracting thofe for the New Moon, and Sun's diftance from the Node, from the faid remainder, and adding thofe for the Moon's Anomaly to it ; and the refult will give the mean time of New Moon In January, the year of the Cen- tury propofed: which being found, D 2 work, [ 2 o] Work, in all refpe&s, for the true time of New or Full Moon in January or any other month of that year, as al- ready fhewn* jV. J5. If the days annexed to the Centuries taken out from Table IV. exceed the number of days from the beginning of January, taken out in the 1 8th Century, add a Lunation and its Anomalies, &c. from Table III to thofe taken out from the 1 8th Century ; and then you can make a fub traction. In all calculations for New or Full Moon, either before or after the 18th Century, the variation numbers anfwer- ing to the Centuries in Table V, rnuft be fubtracled from the mean time of New Moon, and from the Sun's mean diftance from the Node ; and added to the Moon's mean Anomaly, as found for the given time, by the preceding Tables. • Example [ « 5 Example III. For the true time of New Moon in Aprils A, D. 237. From A. D. x 7 3 7 fubtradl: 15 Cen- turies [viz. 1500 years) and there will remain 237. by me Precepts. JNew Moon. Sun's Anom. Moon's Anom. Sun from Node. D. H. M. S. « ' S. 0 ' S. 0 ' Jan. 1737 — 1 500 Years 19 14 58 S 2 3 44 13 15 H 1 34 7 1 58 11 20 27 10 4 59 5 28 2 4 20 14 7 19 39 Remains Var, 1500 Y. 7 11 3 1 4 6 57 + 1 3° 9 0 35 —0 30. Jan. A. D. 237 + 3 L un. »3 13 4 88 14 1 ? 7 11 31 2 27 ig 4 8 27 2 17 27 9 0 5 321 April Firft Equ. i-' 3 5 2 + 3 1 10 8 50 Ar. 1 F j. 0 25 $4 + 1 13 026 Sec. Fqu. 12 7 5 —4 I f O 6 27 7 ' . 2 Eo, True time 12 2 55 ■- ' "■■ ■»*■ -'"J Hence, the true time required is April 12, at 5 5 minutes pail: II m the Afternoon, i v. > To [ 22 ] HI. To calculate the true time of New or Full Moon in any given year and month before the Chrijlian JEra% Old Stile. Find a year in tke i 8th Century, Old Stile, which being added to the given number of years before Chrift, diminifhed by one, fhall make a com- pleat number of Centuries. Find this number of Centuries in Ta- ble IV. andfubtracl: the numbers belong- ing to them from thofe for January, in the $8th Century > and to the remain- ders join the variations for the like num- ber of Centuries from Table V. and then proceed, as above taught, in ap- plying the Equations to gain the true time required. The Moon's motion in her Orbit being now quicker than it was in for- mer ages, is the reafon for our giving the fifth Table, anfwering to her ac- celerations. Example t Example IV. For the true time of New Moon in May y Old Stile, the year before Chrifl 585. The years 584, added to 1716, make 2300 years, or 20 complete Centuries* By the Precepts. New Moon Sun's Anom. Moon's Anom. Sun from Node. D.H.M. S. 0 '* S. 0 ' S. 0 ' Jan. 17 16 — 2300 Years 11 16 6 n 3 39 6 24 25 11 17 18 21238 * 4 43 % 25 54 7 24 11 Remains Var. 2300 Y. O 1*2 27 —3 4° 7. 7 »7 1 7 55 -r-3 3 2 7 1 43. — 1 11 B. Chr. 585 + 5 Lun. 0 8 47 147 15 40 7 7 17 4 2 5 32 1 11 27 4 9 5 7 0 3 2 < 5 3 21 May Firft Equ. 28 0 27 — 12 0 2 49 Ar. 1 Eq. 5 2° 3 2 —5 0 3 53' Sec. Equ. 28 0 15 + 1 31 5 20 27 Ar. 2 Eq. Tr. time May 28 1 46 So the true time was May 28th, at 46 minutes paft 1 o'clock in the Af- ternoon. IV. To i 2 4 3 IV. To calculate the true time of J^ew or Full Moon in any year and month after the iBth Century in the old Stile. Find a year of the fame number in the 1 8th Century with that of the Century propofed, and take out the New Moon and Anomalies for Janu- ary, from Table I. for the faid year in the 1 8th Century : then from Table IV. take out the numbers for the Cen- turies after the i8th, adding them to thole of the 1 8th ; to which join the Centurial variations, and then proceed for the true time of New or Full Moon as fluewn in the former Precepts.] i t ' i i I Example [»S] Example V. Tor the true time of Full Moon in April, Old Stile, A, D. 1903. To A. D. 1703 add 200 years, and the fum will be A. D. 1903. By the Precepts. New Moon. Sun's Anom. Moon's Anom. Sun horn Node. u • jn. ivi. Co 0 Co * Jan. 1703. + |Lun. 6 5 55 14 18 22 6 18 38 0 H 33 7 26 8 6 12 54 6 7 s: 0 15 20 Full D Jan. -f-zoo Years zi 0 iy 8 16 10 7 3 11 0 6 29 292 5 0 37 6 23 17 9 8 48 Var. 200 Years 29 16 27 — 2 7 9 40 7 9 39 + 2 4 2 5 — 1 January + 3 Lun. 29 16 25 88 14 12 7 9 40 2 27 19 7 9 4i 2 17 27 424 3 2 1 April 1903 Firft Equ. 28 6 37 + 3 18 10 6 59 Ar. 1 Eq. 9 2 7 § + 1 15 7 4 5 Second Equ. 28 9 55 -8 55 9 28 23 Ar. 2 Eq. Tr. time, Apr. 28 1 0 Thus the true time is found to be April 28th, at 1 o'clock in the Af- ternoon, E In In calculating forward from A. Da 1 800, the eafieft way is to keep by the Old Stile, and then reduce it to the New, by adding the days difference of Stiles, which will be 1 2, from A. D. 1800 to 1 goo. If the Old and New Stiles had ex- ifted from the beginning, there would have been no difference between them in A. D. 200. But from that time forward there would to the end of the world. And, in order to find always how many days (from the 200th year after Chrift's birth) muft be added to the Old Stile to reduce it to the New, in any given Century, obferve the fol- lowing rule. Divide the number of the given Cen- tury by 4, and ( without regarding the remainder ', when there is any) add 3 to the quotient \ then fubtraEi the fum from the number of the century and the remainder will be the number of days fought. Thus, for the 1 8th Century : which began with A. D. 1701, and will end 1 " ' 1 with [ 2 7\] with A- D. 1800; the fourth part of 18 (omitting fractions) is 4, which added to 3 makes 7 ; and 7 being fub- tra&ed from 18, leaves 11 remaining, for the number of days between the Old and New Stile. Again, for the 1 9th Century, which will begin with A. D. 1801, and end with A. D. 1900, a fourth part of 19 (without regarding fractions) is 4, which being added to 3 makes 7 ; and 7 be- ing fubtra&ed from 19, leaves 12 re- maining for the number of days that muft be added to the Old Stile to re- duce it to the New, from A. D. 1 800 to 1900 ; and fo on. When it appears, by fuch as the foregoing calculations of New and Full Moons, that the Sun's diftance from the [Afcending] Node of the Moon's Orbit is lefs than o Signs 1 8 degrees, or more than 5 Signs 1 2 degrees, fo as not to exceed 6 Signs 1 8 degrees $ or when it is more than 1 1 Signs 1 2 degrees, at the time of New Moon, the Sun will be eclipfed at that time. And E 2 when [-28 ] when the Sun's diftance from the Node is lefs than o Signs 1 2 degrees, or any thing between 5 Signs 1 8 degrees, and 6 Signs 1 2 degrees ; or more than 1 1 Signs 1 8 degrees, at the time of Full Moon, the Moon will be eclipfed at that time. On thefe principles it appears that there muft be Eclipfes at the times mentioned in all the preced- ing Examples except the laft. The reafon of this is, that the Defcendinp- Node of the Moon's Orbit is directly oppofite to the A (tending ; that is, they are jufr. 6 Signs from each other. And when the Sun is within 1 8 degrees of either of the Nodes at the time of New Moon, the Sun will be eclipfed at that time. And when the Sun is within 12 degrees of either of the Nodes at the time of Full Moon, the Moon will then be eclipfed. Becaufe mod people are fatisfied with knowing on what days of the months the Moon is New and Full, without regarding the time of the day, I mall here give a Table of all the days days of the months on which the mean changes df the Moon fall, from A. D. 1752 to 1800, in the New Stile. The days of Full Moons are then eafily found ; for when the Change happens before the 15th day of the month, 15 days added to the day of change, will give the day of full Moon ; and when the Change is after the 1 5 th day of the month, 15 days fubtracled therefrom will give the day of Full Moon. Within the above limits, the day of any month on which the Moon chang- eth, in any given year, is found under that month, and right againft the year. Thus, fuppofe it was required to find on w T hat day of March the Change hap- pens in A. D. 1767 : under March at the head of the Table, and againft 1767 at the left hand is 30 ; the day of the Change required. Where the figures are double, as 3 r 0 or 3 \, againft any year, and under any month; they mew that the Moon changes on the ift day of that month, and alfo on the 30th or 31ft thereof. A Talk [ 3° 3 A Table jhewing on what days of the months the mean Changes of the Moon fall, from A. D. 1752, to A. D. 1800. New Stile. Years. &> s cr M -s fc: June | crq f ^ re -a c p 3 0 <: Dec. 1752 16 15 J 3 12 1 1 IO S 8 26 6 6 4 3 4 3 2 j 3° 28 27 **5 25 31 16 16 1/54 23 22 2 3 22 21 20 '9 18 H 14 12 1 1 1 2 I I I 1 10 9 7 6 5 4 3 j 7 c 6 1 1 29 28 2 7 26 24 23 22 21 3 1 3° 1757 20 18 20 18 18 16 16 H 12 1 1 10 1758 9 8 9 8 7 6 5 4 2 2 3° 29 1 759 28 26 2 3 2 7 26 2 5 24 2 3 21 3 1 21 19 l 9 1760 17 16 16 H *3 12 1 1 9 9 7 7 I76I 0 4 0 4 4 2 2 3° 20 20 2 7 26 I762 3 1 2 S 2 3 2 5 2 3 23 21 21 *9 18 17 16 '5 I763 14 1 2 14 J 3 12 1 1 1 1 9 7 7 5 5 I764 3 2 2 1 30 28 28 27 2C 23 23 1765 3° 16 21 18 21 20 l 9 18 l 7 H 12 12 1766 11 9 1 1 9 9 7 7 5 4 3 2 1 1767 29 28 3° 28 28 26 26 24 23 22 21 20 1768 17 18 16 16 H H »3 1 1 1 1 9 9 1769 7 6 7 6 5 4 3 2 3° 29 28 2 7 26 26 31 1770 2 5 2 5 24 2 3 22 21 19 19 17 17 1771 H 16 M 12 12 10 9 8 26 7 6 1772 5 3 4 2 2 3° 3° 28 27 24 24 1773 2 3 21 2 3 21 21 '9 J 9 17 16 16 H 14 1774 1 2 1 1 1 2 1 1 10 9 8 7 5 4 3 3 J 775 1 2 3° 29 28 27 26 24 24 22 22 31 31 1776 20 19 r 9 18 17 16 16 H 13 12 1 1 10 The [ 3i ] The Table concluded. New Stile. * Years. 3 Feb. Mar. May 1 June I 1 — > t Aug. | CO "H O Nov. | Dec. | 1777 9 9 7 / r r 5 3 2 1 31 3° 29 1778 1779 1780 28 1 7 6 26 16 4 28 17 5 26 16 4 26 IS 3 24 H 2 24 2 3 1 20 I O 29 22 12 3° 21 IO 28 20 IO 28 19 8 26 18 8 26 1781 1782 i783 24 H 3 23 12 2 24 14 3 2 3 1 2 2 22 12 1 3i 19 8 21 IO 29 19 8 27 17 7 26 17 6 2 5 16 5 24 i? 4 23 1784 1785 22 io- 20 9 2 ! I G *9 9 18 7 17 6 16 5 H 3 H 3 12 2 12 2 3 1 1786 1787 1788 29 18 8 28 17 29 *9 7 28 17 5 27 17 5 26 »5 A T 2 5 15 3 24 13 2 31 2 ! I 0 2Q 6 25 »5 3 22 12 3° 22 1 1 29 20 10 28 20 9 27 1789 1790 »79? ' I79 2 1793 1794 1/95 179b 26 1 5 5 23 12 1 20 9 24 '4 3 22 10 J 9 8 26 H 5 22 12 1 3 1 20 8 24 14 3 21 1 0 29 19 7 24 »3 5 21 IO 29 18 7 22 I 2 I 19 8 27 17 5 22 j 1 1 3° g 27 16 5 l 9 8 27 16 c 5 24 13 2 19 8 27 15 r j 2 3 1 31 20 9 28 >7 6 25 '4 3 22 1 1 29 17 6 25 *3 2 22 1 1 29 1797 1798 1799 27 17 6 26 IS 5 27 17 6 26 5 25 15 4 24 13 3 23 2 22 1 1 1 3° 21 IO 29 19 8 27 18 8 26 1 800 2 5 2 3 2 5 2 3 22 21 2C 19 17 1 1 ? * 5 IS This Table begins the day at mid- night, which is according to the com- mon way o f reckoning. _ Look 1 [ 32 ] Look for the given year in the left hand column, and againft it under the given month you have the day of mean New Moon in that month. Of the caufes and times of Eclipfes. An Eclipfe of the Sun is caufed by the Moons opaque body paffing be- tween the Sun and thofe parts of the earth from which me hides the whole or part of the Sun : and this can never happen but at the time of New Moon. An Eclipfe of the Moon is caufed by the whole or part of her body paffing through the earth's fhadow : which can never happen but when the Moon is full. If the Moon's Orbit lay in the plane of the Ecliptic (in which the Earth al- ways moves, and the Sun appears to move) the Sun would be eclipfed at the time of every New Moon ; and the Moon would be eclipfed at the time of every Full. But [ 33 ] But one half of the Moon's Orbit lies on the North fide of the Ecliptic, and the other half on the South fide of it. Therefore the Moon's Orbit interfedts the Ecliptic only in two op- pofite points, which are called the Moons Nodes ; and the angle which the Moon's Orbit makes with the Ecliptic is 5 c 1 8'. The interfe&ion from which the Moon afcends North- ward from the Ecliptic is called the Moon's Afcending Node\ and the op- pofite interferon, from which the Moon defcends Southward from the Ecliptic is called the Moons Defcending Node. Thefe Nodes move backward in the Ecliptic i 9* degrees every year, from the confequent toward the ante- cedent figns, and therefore they go quite round the Ecliptic, in 18 years, 225 days, and 5 hours. From the time of the Sun's being in conjunction with either of the Moon's Nodes, to the time of his being in con- junction with the other, is about 173I days, at a mean rate; within which F number [ 34 ] number cf days the Eclipfes rauft al- ways happen, in different times of the year. Iloe days of thefe Conjun&ions are Wewn in the following Table, from A. D. i j 52 to 1 800, N. S. Mean Conjunctions of the Sun and Nodes. Years. 1752 1753 1754 175? 1756 1757 1758 *759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 *774 1775 1776 Ale. Node. Mon. D. Nov. oa. Sept. Sept. Aug. Aug. July June June May May Apr. Mar. Mar. Feb. Jan. Jan. Dec. Dec. Nov. oa. oa. Sept. Sept. Aug. July May Apr. Apr. Mar. Mar. [an. 29 Jan. Dec. 1 c 18 3 1 1 2 23 5 '7 29 10. 22 4 it 2 Defc. Node. Mon. D. Nov. Nov. 40a. oa. Sept. Aug. Aug. July July June May May Apr. Apr. Mar. Feb. v «'eb. 16 28 9 2 1 2 1 z 25 6 18 3° 1 1 24 6 17 29 1 2 4 1 1 28 »4 ?3 Years. 1777 778 779 780 78i 782 783- 784 785 786 787 788' 789 790 79i 792 793 794 795 796 797 798 799 800 Afc. Node. Mon. D. July June June May Apr. Apr. Mar. Mar. Feb. an. Jan. Dec. Nov. Nov. oa. oa. Sept. Aug. Aug. July July June May May Apr. 10 Dele. Node. Mon. D. Jan. 17 Dec. 30 Dec. 1 1 Nov. 23 1 4 Nov. 4 26 oa. m Sept. 28 Sept. ic Aug. 22 Aug. 1 Ju'y 15 June 27 20 1 to 22 4 16 27 Jun 2 ( 3 M May May Apr. Mar. Mar. Feb. Jan. Jan. Dec. 22 Dec. 4 Nov. 1 oa. 28 oa. 9 9 20 2 «3 2 ; 7 16 28 When [ 35 ] When the Moon changes within 1 8 days before or after the day of the Sun's being in conjunction with either of her Nodes, the Sun will be eclipfed : and when the Moon is full within 1 2 days before or after the day of the Sun's conjunction with either of the Nodes, the Moon will be eclipfed. At greater diftances of the Sun from the Nodes, there can be no Eclipfes of thefe Lumi- naries. As the Table contained on page 30 and 3 1 fhews the days on which the mean changes of the Moon happen, and the Moon is always full on the 1 5 th day before or after the change; and the Table on page 34 mews the days on which the Sun is in conjunction with the Moon's Nodes ; we may ea- fily find by thefe Tables on what days of any given year from A. D. 1 7 5 2 to 1800, the Sun and Moon mud be eclipfed. As, for example. In the year 1766, the Sun is in conjunction with the Moon's Afcending Node on the 1 8 th of February, and F 2 with [ 36 ] with tlie Defcending Node on the i oth of Auguft. Now, I find by the Table, page 30, and 3 1 , that in the year 1766, the changes of the Moon are on Jan. ii, Feb. 9, March 11, April 9. May 9, June 7, July 7, Aug. 5, Sept. 4, 061. 3, Nov. 2, Dec. 1 and 31 ; and confequently, as the change on Feb. 9th is within 18 days of Feb. 1 8th when the Sun is in conjunction with the Afcending Node, the Sun muft be eclipfed at the time of that change. And as the change on Auguft 5 is within 18 days of Auguft 10, when the Sun is in conjunction with the Defending Node of the Moon's Orbit, the Sun muft be eclipfed at the time of that change alfo. But as all the other changes of the Moon in that year are more than 18 days from the times of the conjunction of the Sun and Nodes, there can be no more than the two abovementioned Eclipfes of the Sun in the year 1766. By adding 15 days to all the chan- ges of the Moon ? in the fame year, we C 37 ] we find the days of all the Full Moons to he Jan. 26, Feb. 24, March 26, April 24, May 24, June 22, July 22, Auguft 20, Sept. 19, Oct. 18, Nov. 17, and Dec. 16. But of all thefe Ful Moons, there are only two which happen within 1 2 days of the conjunc- tions of the Sun and Nodes ; viz. thofe on the 24th of February and 20th of Auguft : and therefore, it is only on thefe two days of the year 1766, that the Moon can be eclipfed. And thus we have a very plain and eafy method for finding how many Eclipfes there muft be of the Sun and Moon in any given year, and the days on which they muft fall, according to the mean times of New and Full Moons, from A. D. 1752 to A. D. 1800. But to fhew how to calculate the true times and places of Eclipfes for different parts of the earth, would fwell out this volume far beyond the intended bulk : and therefore, for fuch calculations and projections, I beg leave to refer the curious reader to my fyftem of [38] of Aftronomy, printed for Mr. Millar, Bookfeller in the Strand, London; to be now had of Mr. Cadell, fucceflbr to Mr, Millar, at his fhop oppofite Ca- therine ftreet in the Strand. The following Table Jhews the Suns true place in the Ecliptic^ and his decli- nation from the Equator , at the noon of every day of the fecond year after Leap year^ on the meridia?i of Greenwich, The figns of the Ecliptic are marked in the Table as follows. Aries f, Taurus Gemini H, Cancer og, Leo £t, Virgo n%, Libra Scorpio vt\, Sagittarius £ , Capri- ccrnus fcf, Aquarius z? t) and Pifces >£• A Table of the Sun's Declination is very ufeful for finding the Latitudes of places on the Earth. And as the method of doing this by the Declination of the Sun is generally known, we have given the following Table for that pur- pofe, to the neareft mean between Leap year, and the firft, fecond, and third year after. A Table [ 39 3 A Table Jhewing the Suns Place and Declination, January. February. Sun's PI. S.'s Dec. Sun's PI.. S 's Dec : S. 0 ' O * s. ° r 0 ' — I T^C I T 1 1 5 S.23 1 3°. 0. 17 3j. z 1 2 6 22 5 6 1 3 39 10 45; 3 1 2 8 22 5° I A 4 U I 6 2 q 4 H 9 22 44 T f 4 1 1 6 I o : 5 t r 1 5 1 0 2 2 37 16 4 1 -15 52; 6 1 u 1 1 22 3° l 7 .42 1 5 33 7 / T H l 7 1 2 22 2 2 j g A h 43 1 5 K 8 I 0 13 22 J 9 43 14 50 9 '9 '4 22 0 20 44 14 36 io 2 O 16 21 57 ^ t 45 14 17 1 1 2 I 1 1 1 / 2 1 48 2 2 4S l 3 57 12 % 2 1 8 2 1 3 8 ■* j 4 U 13 37 '3 2 3 l 9 2 1 25 24 46 *3 l 7 14 24 20 2 1 17 2 5 47 12 57 15 z 5 2 1 21 6 26 47 12 37 16 26 22 20 55 27 48 12 16 17 27 24 20 43 28 48 11 55 18 28 2 5 20 3* 29 48 11 34 19 29 26 20 18 X 0 49 1 1 12 20 £.* 0 2 7 20 5 I 49 10 51 2 1 1 zg 19 5 2 2 50 10 29 22 2 29 19 39 3 5° 10 7 2 3 3 3° 19 2 4 4 5° 9 45 24 4 3i 19 1 1 5 5i 9 23 2 5 5, 32 18 55 6 5 1 9 1 26 0 33 18 40 7 5 1 9 38 7 34 ! 18 2 5 8 5i 8 16 :8 8 35 | 18 9 9 5i 8 53 20 9 53 3 5 36 10 i '7 37 N. fig nifies North Decl. ' £1 1 1 37 ! 17 20 and S. South Decl. 7 0 47 N. 0 *9 1 1 2 1 1 55 ■ 22 I 46 0 42 2 1 1 12 *5 2 3 2 46 1 6 3 9 12 35 24 3 45 1 3° 4 7 12 55 25 4 44 1 11 5 6 13 *S 26 5 43 2 6 4 13 34 27 6 43 2 40 7 2 13 54 28 7 42 3 4 8 0 H 1 2 29 8 4 1 3 27 8 59 3' 30 9 40 3 5° 9 57 5° 3i 10 39 4. 13 Look for the month at the top of the Table, and under it againft the given day of the month, you have the Sun's [ 4* 3 The Table continued. May. June. Sun's PI. S.'s Dec. Sun's PI. S.'s Dec. S. 0 ' 0 S. 0 0 ' — I « 10 55 N.15 8 26 n 10 44 N.22 6 2 1 1 53 15 1 1 41 22 14 3 12 5 1 *5 43 12 39 22 21 4 13 49 16 1 13 36 22 28 5 H 47 16 18 14 34 22 35 6 15 45 16 35 15 3 l 22 42 7 16 43 16 52 16 28 22 47 8 17 4 1 l 7 8 17 26 22 53 9 18 39 17 24 18 23 22 5 8 IO 19 36 17 40 19 2C 23 3 1 1 20 34 *7 55 20 l8 •? *p 8 12 21 32 18 1 1 21 <5 23 12 i3 22 30 18 26 22 12 23 15 H 23 28 18 40 23 9 23 18 1 5 24 25 18 55 24 7 23 21 16 25 23 x 9 9 25 26 4 23 23 17 26 21 19 22 1 23 25 18 37 *9 & 36 26 58 23 26 r 9 28 16 49 27 56 23 28 20 29 14 20 1 28 53 23 29 21 n 0 1 1 20 H 29 5° 23 29 22 1 9 20 26 35 0 47 23 29 23 2 7 20 37 1 45 23 28 24 3 4 20 49 2 42 23 27 4 2 21 0 3 39 23 26 26 4 59 21 10 4 3 6 23 24 27 5 57 21 20 5 33 23 22 28 6 54 21 30 6 3 1 23 19 29 7 5 2 21 40 7 28 23 16 3 C 8 49 21 49 8 25 23 13 3i 9 47 21 58 Sun's Place in the Ecliptic, and his De- clination as it is then North or South. By means of this Table, and the Table of the Semidiurnal Arcs of the G Sun [ 42 ] Table continued. July. Auguft. •< Sun's Pi. S.'s Dec. Sun's PI. S.'s Dec. S. 0 ' 0 ' S. * ' 0 ' I ?E 9 22 N.23 0 0 8 5 8 NT 18 3 2 1 0 l 9 2 X 3 c > 9 55 1 fj v ^ 8 48 3 1 1 I u 2 3 j 0 I u 53 I -7 3 2 4 1 2 J 4 22 r r 5 3 I I 5° 1 7 1 0 5 1 3 I I 22 CO J I 2 4» */ 0 u H Q O 22 44 13 45 1 u 44 7 ! 5 5 2 2 l8 *4 43 1 LI 27 8 ID 1 22 a 1 1 5 4 1 16 1 0 9 1 ft 1 O 59 2 2 22. 1 0 3 s 1 > 53 1 0 1 7 5 5 22 1 7 -?ft 3° T C l S 35 1 1 1 Q I 0 54 22 9 IS 33 18 I 2 *9 5' 22 j 3 1 t r *5 0 13 20 49 2 1 20 29 4 1 »4 2 I 4(3 2 1 43 2 1 zu 2 3 I s 2 2 43 21 34 2 2 24 I A \T 4 I O 23 40 2 1 24 23 22 t 1 *3 45 !7 24 38 21 14 24 20 J 3 26 I 0 2? 35 21 4 2 5 17 13 7 19 26 32 20 53 26 15 1 2 47 20 27 29 20 42 27 13 12 28 2 I 23 2 7 20 3° 28 I I 12 8 22 29 24 20 *9 29 9 1 1 47 2 3 £3 0 21 20 7 7 1 1 2 7 24 1 19 19 54 1 5 11 7 2C j 2 16 19 4 1 2 3 10 46 3 13 19 28 3 1 10 2 5 27 4 1 1 19 «5 3 59 10 4 28 5 8 19 1 4 57 9 43 29 6 6 18 47 5 55 9 22 30 7 3 18 33 6 53 9 0 ^1 8 0 18 18 7 5i 8 38 Sun and Moon, the times of rifing and fetting of the Sun, on any day of the year, may be found, in all latitudes from 48 degrees to 59 inclufive. The Sun's [ 43 ] The Table continued. September. Oflober. to Sun's PI. S.'s Dec. Sun's PI. S.'s Dec. — S. 0 ' 0 S. 0 I W 8 49 N. 8 '7 £S 8 8 S. , H 2 9 47 7 55 9 7 3 37 3 10 46 7 32 10 7 4 1 4 11 44 7 10 1 1 6 4 24 5 12 4 2 6 48 12 5 4 47 6 13 40 6 26 '3 4 5 10 7 H 39 6 3 H 4 5 33 8 *S 37 5 4 1 *5 3 5 56 9 16 3S 5 18 16 3 6 20 IO 17 34 4 55 17 2 6 42 ii 18 3 2 4 32 18 1 7 5 12 '9 3 1 4 9 19 1 7 28 •3 20 29 3 47 20 0 7 5° H 21 28 3 2 3 21 0 8 13 15 22 26 3 0 22 0 8 35 16 2 3 2 5 2 37 23 0 8 57 »7 24 24 2 H 23 59 9 19 18 2 5 22 1 5 1 24 59 9 4 1 *9 26 21 1 27 25 59 10 3 20 27 20 1 4 26 58 10 25 21 28 19 0 40 2 7 58 10 46 22 29 17 Q 17 28 58 1 1 8 23 0 16 S. O 7 29 58 1 1 29 24 I 15 O 3° ni 0 58 1 1 5° 2 5 2 H O 53 1 58 1 2 1 1 26 3 13 I 17 2 58 12 31 27 4 12 I 40 3 58 12 52 28 5 1 1 2 4 4 58- 13 12 29 6 19 2 27 I 58 13 32 30 7 9 2 51 58 13 52 31 7 58 '4 1 1 Sun's Declination is alfo ufeful for find- ing the Latitudes of Places ; for which I have given a great variety of Rules in my Book of Lectures on Mechanics, G 2 Hydro- [ 44 ] The Table concluded. November. December. Days Sun's PI. S.'s Dec, Sun's PI. S.'s Dec. «— S. 0 > 6 S. 0 ' Q ' I m, 8 53 S. 14 3i $ 9 16 S.2I 53 2 9 5 8 H 5° 10 *7 22 2 3 xo 59 9 1 1 18 22 1 1 4 1 1 59 15 2tf 1 2 "9 22 l 9 5 i? 59 '5 46 J 3 20 22 27 6 »i 59 16 4 22 22 34 7 15 0 16 22 16 23 22 41 8 16 0 16 39 24 22 47 9 17 0 16 57 ?7 25 22 53 10 18 i ! 7 18 26 22 59 1 1 ! 9 1 *7 3° 1 9 27 2 3 4 12 20 2 17 47 20 28 - 2 3 8 *3 21 2 18 3 21 29 23 12 *4 22 3 18 *9 22 3° 2 3 r(5 15 2 3 4 18 34 2 3 3 1 23 19, 16 24 4 18 49 2 4 3 2 23 22 *7 2 S 5 J 9 4 25 33 23 25 18 26 5 19 26 34 23 21 19 27 6 H 33 27 35 23 2$ 20 28 7 19 47 28 36 23 29 21 29 7 20 0 29 37 23 29 2 2 8 20 13 y? 0 38 23 29 2 3 1 9 20 26 1 39 2 3 28 24 10 20 33 2 40 23 27 2 5 3 1 1 20 5° 3 4 2 2 3 26 26 4 1 1 21 1 4 43 23 24 3 7 5 1 2 21 12 5 44 23 21 28 6 H 21 23 45 23 T 9 29 ■7 14 21 33 7 46 23 3° 8 15 21 43 8 48 23 1 1 ? r . 9 49 23 7 Hydroftatics, Pneumatics, and Optics 1 printed for Mr, Cadell, Bookfeller ia the Strand* London. £ 45 ] The Table of Semi-diurnal Arcs for Jhewing the times of rijing and Jetting of the Sun, and Moon, Deg. i 2 3 4 5 6 7 8 9 10 1 1 12 14 16 20 21 22 23 Latitude of the Place. Sun Moon H.M. 6 34 6 39 IB i,9 z 3 28 7 34 7 39 H. 6 6 6 6 6 6 43 6 48 6 53 57 1 6 12 '7 22 27 7 3? 7 ■ 37 7 43 7 49 7 54 8 o 8 6 8 12 8 ic 49* Sun Moon H.M H.M. 12 17 21 26 31 36 41 6 45 6 50 6 55 7 o 7' 5 7 10 7 IS 7 21 7 26 '7 3i 7 37 7 43 7 49 7 55 8 1 8 4 6 20 6 2/1. 6 29 6 33 H.M. 6 8 6 13 6 18 22 7 24 3 1 36 4 1 46 S 2 58 8 5 8 ji 8 17 8 20 Sun Moon 6 57 7 2 7 29 35 4* 47 53 59 ■8 6 8 9 H.M 8 9 8 15 8 22 8 2; ^ jfe^/ time of Sun-rifing and Sun- Jetting) on any given day of the year, [46] The Table of Semi-diurnal Jlrcs continued. Latitude of the Place. 48 ( 49' Sun IMoon Sun Moon Sun Moon H. M. Deo- H.M. H.M. 11 . IVi. H.M. H. M. 5 59 6 11 5 59 6 T T 1 I 5 59 u 3 1 2 5 54 O 0 5 54 6 6 5 54 0 /; 0 3 5 5° 6 1 5 49 6 ] 5 49 0 I 4 5 45 5 57 5 45 5 57 5 44 5 5 r J t 5 4 1 5 53 5 4° 5 52 5 39 5 5 1 5- 3° 5 4 6 5 35 5 47 5 35 5 46 7 5 32 5 44 5 3i 5 43 5 30 5 42 Q o 5 2 7 5 3* 5 26 5 37 s 25 5 36 9 5 2 3 5 33 5 2 » 5 3i 5 20 5 5° IO 5 18 5 28 5 »7 5 27 5 15 5 2 5 1 1 5 13 5 23 5 J 2 5 22 5 10 5 20 12 5 9 5 »9 5 7 5 17 5 5 5 »5 *3 5 4 5 H 5 ? 5 12 5 0 5 1 0 l 4 4 59 5 9 4 57 5 7 4 54 5 4 *5 4 54 5 4 4 52 5 2 4 49 4 59 16 4 49 4 58 4 46 4 56 4 45 4 54 ! 7 4 44 4 53 4 4» 4 51 4 38 4 48 18 4 39 4 48 4 36 4 46 4 33 4 42 19 4 34 4 43 4 39 4 40 4 27 4 36 20 4 28 4 37 4 25 4 33 4 21 4 29 21 4 23 4 3' 4 *9 4 27 4 »5 4 2 3 22 4 17 4 25 4 »3 4 21 4 9 4 *7 23 4 11 4 T 9 4 7 4 15 4 3 4 1 ! 1 1 4 8 4 16 4 4 4 12 4 0 4 8 50 s £fi 6 5 10 5 4 4 58 4 5 2 4 47 H.M 6 4 5 59 5 53 5 48 5 4 2 5 37 5 3i 5 20 5 21 5 14 5 8 41 34 28 22 *j 9 2 5 5 47 3 43 4 44 38 3' 24 17 10 3 57 54 you have the Sun's Semi-diurnal Arc, or time of his fetting, on that day ; which Arc being doubled, gives the whole length of the day; and being fubtraded from 12, gives the time of Sun-rifing. Thus, foppofe the Lati- o tude t 49 ] The Table of Semi-diurnal Arcs continued. Lat itude of the PI ace. o 5' 54 55 Q 56° c bun Moon bun Moon Sun Moon Deg. 0. M. TT ft/1 H. IVJ. rr i\/T £1. 1V1. ri. iVJ. tl. M. M. IVJ. ■ c o i /- 1 /C >. J 0 21 A 1 6 9 A _ , A * n O 10 A - » 0 22 2 6 15 A _ a 0 20 O 27 0 10 A O 1 0 28 1 re 3 6 20 6 32 6 21 6 33 A O 22 6 34 c w 4 6 26 6 38 6 27 6 39 6 28 6 4 o s 5 6 31 >* 6 44 6 32 6 45 I 34 6 47 6 6 3? 6 5 1 6 38 6 52 0 40 6 54 -s 7 6 43 a _ 6 57 a . . 6 44 6 58 K A 0 46 7 c 8 6 48 7 2 6 50 7 4 a 0 52 7 6 "O" 9 6 54 7 8 7 0 6 56 7 10 6 58 7 12 c c ?o 7 0 7 H 7 2 7 16 7 5 7 19 g 1 1 7 6 7 20 7 8 7 22 7 u 7 25 12 7 12 7 26 7 15 7 29 7 18 7 3* ■ u Cr 13 7 18 7 3 2 7 2i 7 35 7 24 7 38 H 7 24 7 37 7 28 7 4 2 7 '3* 7 4 6 J.fj 7 3i 7 45 7 34 7 49 7 39 7 53 O 2. 16 7 37 7 5 2 7 4i 7 5 6 7 45 8 G 17 7 44 7 59 7 48 8 3 7 5 2 8 7 18 7 5' 8 7 7 55 8 10 8 0 8 15. *9 7 58 8 14 8 2 8 18 8 7 8 23 20 8 5 8 21 8 10 8 26 8 15 8 31 21 8 12 8 28 8 18 8 34 8 24 8 40 22 8 20 8 36 8 26 8 42 8 3 2 8 48 2 3 8 28 8 44 8 34 8 50 8 41 8 57 ; 8 32 8.49 8 38 8 54 8 46 9 1 : tude to be 52 degrees North * and the day to be the 4th of May, when the Sun's Declination is 16 degrees North. Then, under 52 °at the head of this Table, and againft 1 6 degrees of North H decli- [ So] The Table of Semi- diurnal Jlrcs . continued* Latitude of the Place. o P* 54° 55° 56° I Sun — i Moon Sun Moon Sun Moon Deg. H. M. H.M. H.M. H M. H. M. H. M. u I - 5 58 — 6 10 5 58 6 10 5 58 6 10 O o 2 5 53 6 5 5 53 6 5 5 5 2 6 4 3 5 47 5 59 5 47 5 59 5 4 6 5 58 c C3 4 5 42 5 54 5 41 5 53 5 4® 5 52 5 5 3 6 5 47 5 35 5 46 5 34 5 45 6 5 3° 5 4i 5 29 5 4° 5 28 5 39 4) •jC 7 5 25 5 36 5 23 5 34 ■5 22 5 33 Cn 8 5 »9 5 3° 5 17 5 28 •5 16 5 27 ' o 9 5 13 5 23 5 12 5 22 5 10 5 20 c o IO 5 8 5 18 5 6 5 17 5 3 5 H 1 OB 1 1 5 2 5 12 4 59 5 »° 4 57 S 8 C 12 4 5 6 5 6 4 53 5 3 4 5 1 5 1 "o J 3 4 5° 5 0 4 47 4 56 4 44 4 53 Q 14 4 44 4 5? 4 41 4 50 4 37 4 4 6 3 15 4 37 4 48 4 34 4 43 4 3i 4 40 c 16 4 3» 4 4i 4 27 4 36 4 24 4 33 17 4 z3 4 34 4 21 4 29 4 17 4 25 18 4 18 4 26 4 14 4 22 4 9 4 17 19 4 »i 4 -9 4 7 4 15 4 2 4 10 20 4 4 4 12 3 59 4 7 3 54 4 2 21 3 57 4 5 3 52 3 59 3 4 6 3 54 22 3 5° 3 58 3 44 3 5 2 3 38 3 45 23 3 42 3 49 3 3 6 3 44 3 29 3 36 1 3. 3 38 3 44- 3 32 3 4° 3 25 3 3 2 declination, I find 7 hours 30 minutes to be the Sun's femi- diurnal arc on that day ; which being doubled gives 1 5 hours for the whole length of the day. The feid arc (hews that the Sun fets C si ] The Table of Semi-diurnal Arcs continued. M Lati tude of the Pla ce. ■ o 5" 57° 58 0 59° Sun Moon _____ Sun • Moon ______ Sun ' Ml 1|- Moon ueg. H.M. H.M. - H.M. H.M. H.M. H.M. c o I 6 io 6 22 6 10 ___— _ 6 22 6 11 6 23 o 2 6 16 6 28 6 17 6 29 6 17 6 30 s •id 3 6 22 6 34 6 23 6 35 6 24 6 37 4 6 2Q 6 41 6 30 6 42 6 31 6 44 C 6 2C 6 48 6 36 6 49 6 38 6 51 V3 6 41 6 55 ^ 43 6 56 6 44 6 58 _t _C 7 6 48 7 2 6 49 7 0 6 51 7 5 Q a 6 54 7 8 6 56 7 9 6 58 7 12 o 9 7 1 7 15 7 3 7 16 7 5 7 »9 c o IO 7 7 7 21 7 10 7 2 4 7 13 7 27 1 1 7 H 7 28 7 J 7 7 3 1 7 20 7 34 c I 2 7 21 7 35 7 2 4 7 39 7 2 7 7 4 2 *3 7 28 7 4 2 7 3i 7 4 6 7 3? 7 5° Q 14. 7 35 7 5° 7 39 7 54 7 43 7 58 _3 }5 7 4 2 7 57 7 4 6 8 2 7 5 1 8 6 o 2 16 7 49 8 4 7 54 8 10 7 59 8 15 J7 7 57 8 12 8 2 8 18 8 7 8 23 18 8 5 8 21 8 10 8 26 8 16 8 32 »9 8 13 8 29 8 19 8 35 8 25 8 42 20 8 21 8 38 8 28 8 45 8 35 8 52 21 8 30 8 47 8 37 8 55 8 45 9 2 22 8 39 8 56 8 47 8 5 8 55 9 '3 23 8 49 9 6 8 57 9 »l 9 6 9 24 I ' "i 8 54 9 i 1 9 2 9 2 9 9 11 q 29 fets at 30 minutes after 7 ; and being fubtraded from 12, leaves 4 hours 30 minutes, which {hews that the Sun rifes at 30 minutes after 4 o'clock. Jtx H 2 this C 5* 3 Tie fable of Semi-diurnal Arcs Q Deg concluded. Latitude of the Place. Sun H.M. 57° Moon H.M. I 5 S 8 610 2 5 52 6 3 3 5 45 j j 5 39 5 Qu. How much time? Lun. 10 Days. Decimals of a day, 295.3059085108 mult, by 24b; 12236340432 6118170216 Hou. 7.3418042592 mult, by 6o m » Min. 20.5082575520 mult, by 60s Sec. 30.4954531200 mult, by 6oth« Th. 29.4271872000 d. h. m. s. th. 'jbifaier 295 7 20 30 29.8. I 2 Example Example II. 74212 mean Lunations Qu. How many days^ hours , minutes y &c. ? Lun. j Days. Decimals of a day. 70000 2.67141.3595756 4000 J 1 1812.2. 36340432 200 10 2 5906.1 18170216 295.3059085108 59 061 18170216 74212 Lun. 21915 24.20824034896 Days, mult, by 24k. contain days. 83296139584 4164 069792 --> Min. 59.86610250240 mult, by 6os« c a 2 ^ «^ a O H3 O Sec. 5 1 .96615014400 mult, by 6oth» 0 0 >s PQ thirds 57.96900864000 y. d, h. m. s. tb. dnfixtr 6000 24 4 59 51 57. 969. Example III; 100000000000 mean Lunations y Qu. .f/w much time f Lun. Days, loocoococooo I 2953059085108 Anfwer. In [ 6« ] In Example III, the number of cyphers annexed to i are equal to the number of decimal parts in the firft line of the Table ; and therefore the whole of that line becomes a whole num- ber of integral days, without any fraction. So that, in 1 00,000,000,000 mean Lunations, there are juft 2953059085108 days. It is fomewhat remarkable, that every 49th mean New Moon falls but 1 min. 30 fee. 34 thirds fhort of the fame time of the day as before. Lun. 40 9 49 Lun. Example IV. Days. Decimals of a day. 1181.2236340432 265.77531765972 1446.99895170292 Days. mult, by 24*1* 399580681168 199790340584 |* 4- Hou. 23.97484087008 mult, by 60m. c o Min. 58.49045220480 ^ mult, by 6os« Sec. 29.42713228800 mult, by 6c th « Thirds. 25,62793728000 ATabk I 62 ] A "Table fiewing how many mean Lu- nations are contained in any given quantity of time. Decimals of a Lunation. 12.368530038627 21.737000077255 37.IO559OI I q882 49.474120154.510 61.842650193 137 74.21 1 180231765 86.579710270392 98.948240309020 III. 316770347647 Decimals of a Lunation. 00.033863189760 00.067726379520 00.101589659280 00.135452759040 00.169315948800 00 203179138560 00.237042328320 00.2709055 1 8080 00.304768707840 £ Decimals of ? a Lunation. 0.0014109662 0.002821934 5 0.0042328987 0.0056438649 0.0070548312 0.0084657974 0.0098767637 0.01 12S77299 0.0126986962 ^ Decimals of ? a Lunation. 0.00,00235 161 0.0000470322 .0000705483 0.0000940644 0.000 1 1*7 5-805 0.0001 410966 0.0001 646 1 27 0.0001881288 0.00021 16449 0^000000391 9 0.0000007838 0.000001 1758 0.00000 1 5677 0.0000019597 0,0000023 5 1 6 0.0000:27435 0.000003 1 35 5 c. 0000035 274 Decimals of ? a Lunation £ Decimals of ? a Lunation. 0.0000000065 0.00000001 31 0.0000000196 0.0000000262 0.0000000327 0.0000000392 0.0000000457 0.0000000522 0.0000000587 For tens of Julian years, days, hours, 8cc. remove the decimal points one place forward ; for hundreds, two pla- ces ; for thoufands, three places ; for tens of thoufands, four places ; and fo on, as in the following example. It appears by the firft line of the above Table, [ 6 3 ] Table, that in iooooo Julian years (which contain 365 25000 days) there are 1236853 mean Lunations, and .0038627, or ji^iss parts of a Luna- tion, which final! fraction may be ne- glected. [In common working, 'tis fufficient to take in only four or jive of the decimal figures.] Example V, In 6000 Julian years, 24 days, 4 hours, 59 minutes, 52 feconds, Qu. How many mean Lunations f Years 6000 **• Hours 4 Min. J 5° t 9 Anfwer, Lun. Decimals. 7421 1.180231765 » 0.677263795 °- I 3S43 2 7S9 0.005643864 More Examples would be 0.001 175805 fuperfluous, 0.00021 1645 0.000019597 G.ooocoo/84 7421 2.000000014 To explain the Phenomena of the Har- veft-Moon, by means of a common globe. Make chalk-marks all round the globe on the Ecliptic, at i z\ degrees from t6 4 ] from each other (beginning at Caprn corn) which is equal to the Moon's mean motion from the Sun from day to day, near enough for your purpofe* Then elevate the North pole of the globe to the latitude of any place in Europe ; fuppofe London, of which the latitude is 5 1| degrees North. This done, turn the ball of the priobe round we ft ward, in the frame thereof ; and you will fee that different parts of the Ecliptic make very different angles with the horizon, as thefe parts rife in the Eaft : and therefore, that in equal times, unequal portions of the Ecliptic will rife. About Pifces and Aries fe- ven of the marks will rife in about two hours and an half, meafured by the motion of the index on the ho- rary circle; but about the oppofite figns, Leo and Virgo, the index will go over eight hours in the time that 7 marks will rife. The inter- mediate figns will, more or lefs, par- take of thefe differences, as they are more or lefs remote from them. Hence [ 6 5 ] Hence it is plain, that when the Moon is in Pifces and Aries, the differ- ence of her rifing will be no more than two hours and an half in feven days ; but in Virgo and Libra it will be eight hours in feven days ; and this happens in every lunation. The Moon is always oppofite to the Sun when me is full ; and the Sun is never in Virgo and Libra but in our Harveft-months, and therefore the Moon is never full in Pifces and Aries but in thefe months. And confequent- ly, when the Moon is about her full in harveft, me rifes with lefs difference of time, for a week, than when (he is full in any other month of the year. Here we confider the Moon as mo- ving always in the Ecliptic. But as fhe moves in an orbit which is inclined to the Ecliptic, her riling when about the full in Harveft will fometimes not differ above an hour and 40 minutes through the whole of 7 days; and, at other times, it will differ three hours* and an half, in a week, according to K the [66] the different portions of the Nodes of her orbit in the Ecliptic, in differ- ent years. In our Winter, the Moon is in Pifces and Aries, about the time of her firft quarter ; and rifes about noon : but her riling is not then taken notice of, be* caufe the Sun is above the Horizon. In Spring, the Moon is in Pifces and Aries, about the time of her change ; and then, as (he gives no light, her riling cannot be perceived. In Summer, the Moon is in Pifces and Aries about her third quarter ; and then, as fhe rifes not till about midnight, her riling pafles unobferved ; efpecially as (he is fo much on the decreafe. But in harveft, Pifces and Aries are oppofite to the Sun ; and therefore the Moon is full in them at that time, and rifes nearly after Sunfet for feveral evenings together; which makes her riling very confpicuous at that time of the year, as it is fo beneficial to the farmers, in affording them an imme- diate C 67 ] diate fupply of light after the going down of the Sun, when they are reap- ing the fruits of the earth. Rules for fqlving Aflronomical Problems by the Logarithmic Tables of Sines and 1 . The Suns Longitude, or diflance from the nearefl EquinoElial point ( viz. the beginning of Aries or Libra) being given \ to find his Declination, As Radius is to the Sine of the Sun's diftance from the neareft Equinoctial point, fo is the Sine of his greateft decli- nation (23° 29') to the Sine of his de- clination fought. 2. The Sims Declination being given, to find his diflance from the nearefl EquinoElial point, and confequently his place in the Ecliptic. As the Sine of the Sun's greateft declination is to the Sine of his prefent Iv 2 decli- [ 68 ] declination, fo is Radius to the Sine of his diftance gone from, or in going toward, the Equinoctial point required. 3. The Suns diftance from the near eft Equino&ial point being given , to find his right Ajcenfion. As Radius is to the Co-fine of the Sun's greateft declination, fo is the Tan- gent of his diftance from Aries or Li- bra, to the Tangent pf his right afcen- fion therefrom. 4. The Latitude of the place, and the Suns Declination being given, to find his Afce7 / ijional difference. As to the Go-tangent of the latitude is to the Tangent of the Sun ! s declina- tion, fo is Radius to the Sine of his afcenfional difference required. 5. The [ «9 ] 5. "The Latitude of the place and the Suns Declination beinggiven, to find his Amplitude, or the number of de- grees he rifes and fets from the Eafi a?id Weft. As the Co-fine of the latitude is to the Sine of the Sun's declination, fo is the Sine of his diftance from Aries or Libra to the Sine of his Amplitude. 6, "The Suns right Afcenfion and 1 is greatefl Declination being given, to find the Angle of the Ecliptic and Meridian. As Radius is to the Sine of the Sun's .greateft declination, fo is the Co -fine .of his right afcenfion tp the Co- fine of the angle fought. 7. The [ 7° ] j. The Latitude of the place and the Suns Declination being given, to find the Sun s Altitude when he is due Eajl or JVejl. As the Sine of the latitude is to the Sine of the Sun's declination, fo is Radius to the Sine of his altitude when due Eaft or Weft. JV. B. By this problem, a true meridian line may be drawn in Summer, when the Sun rifes before he comes to the Eaft, and paries by the Weft before he fets. For, if a long up- right wire be fet in a truly level board, the ftiadow of the wire will run We ft ward on the board when the Sun is due Eaft, and Eaft ward when the Sun is due Weft ; which will be at the inftant when his al- titude, obferved by a quadrant, agrees with what the problem makes it. And then, if two points are marked in the line of the fhadow, and a 7 flraight [ 7* 3 ftraight line be drawn through them on the board, and this line be croffed at right angles, in any point, by another ftraight line, that line will be a true meridian line ; and if the wire be placed perpendicularly in the interferon of thefe two lines, the ftiadow of the wire will cover the meridian line when the Sun is on the meridian of the place. This may de done beft of all about the Summer folftice, becaufe the Sun changes his altitude fafteft, and his declination floweft, about that time. 8. The Latitude of the place and the Suns Declination being given > to find the Suns Altitude at fix d clock in the Morning or Evening, As Radius is to the Sine of the Sun's declination, fo is the Sine of the lati- tude to the Sine of the Sun's altitude at fix o'clock. By this problem, you may know when it is exaclly fix o'clock by the Sun, C 72 ] Sun* and confequently how to place a Sun dial true at that inftant, provided it be done in the Summer-time, when the Sun is above the horizon at fix. For, if you keep watching, and ob- ferving the Sun's altitude with a qua- drant, when you judge the time tar be a little before fix, till you find his altitude agrees with what the problem makes it, you are fure that it is then precifdv fix o'clock by the Sun ; to which time, fet your watch, and then you may fet it to the true equal time by a common Equation table, which fhews how much the Sun's time is fafter or flower than the equal time, every day of the year. IV. B. In all obfervations of the Sun's altitude, you muft fubtrad: the re- fra<&ion of the Sun's rays by the Atrrofphere from the obferved titude ; for otherwife you will not have it true. And for this purpofe, I fhall fubjoin a Table of refra&ions at the end of thefe problems, to C 73 ] &ew how much lefs the true allttude is, than the obferved altitude;- and when it is fo much lefs than the pro- blem gives, as is equal to the quantity of refraction at the time of the obferv- ed altitude, you have the altitude true* And here, with regard to the pla- cing of Sun-dials, I mull make art ob- fervation, that may perhaps feem a very odd one to moft people; which is, that if the Dial be made according to the ftricl rules of calculation, and be truly fet at the inftant when the Sun's center is on the Meridian ; it: will be a minute too faft in the Kortn^oon, and a minute too flow in the Afternioon, by the fhadow of the Stile; for', the edge of the fhadow that {hews the time is even with the Suns foremof edge all the time before Noon, anc even with his hindmoft edge all the Af- ternoon on the Dial. And itiss the Sun's center that determines tfc time in the (fuppofed) Hour circles >ff the heaven. And as the. Sun is haliai de- gree in breadth, he takes two nirnutes L to [74] to move through a fpace equal to his breadth; fo that there will be two minutes at Noon in which the fhadow will have no motion at all on the Dial. Confequently, if the Dial be' fet true by the Sun in the Forenoon, it will be two minutes too flow in the Afternoon; and if it be fet true in the Afternoon, it will be two minutes too faft in the Forenoon* The only way that I know of to re- medy this, is to fet every hour and minute divifion on the Dial one minute nearer XII than the calculation makes it to be. * The Sun moves 1 5 degrees in one hour, 30 degrees in two, 45 in three, 60 in four, 75 in five hours, and 90 in fix, with refpecl to the equator ; but 5 in an oblique fphere, the motion of the fhadow, either on a horizontal or vertical plane, is very different. To find the degrees and minutes of a de- gree of the hour diftances from XII on a horizontal Dial, fay, as Radius is to the fine of the latitude of the place (which [ 75 ] (which is the fame as the angle of the ftiles height) fo is the Tangent of 15 degrees, and of 30, and 45, 60, 75, and 90, to the Tangent of the diftance (in degrees and minutes) of XI and I, X and IT, IX and III, VIII and IV, VII and V, VI and VI, from XII on the Dial. The fame calculation ferves for erect South Dials, only ufing the Co-latitude inftead of the latitude, for the hours and height of the ftile. 9. The Latitude of the place and the Suns Declinatioit being given y to find the Suns Azimuth from the North at fx 0 clock. As the Co-fine of the latitude is to Radius, fo is the Co- tangent of the Sun's declination to the Tangent of his Azimuth from the North at fix. 10. The [ 76 ] 10, "The Suns Altitude, Declination, and time of the day, being given ; to find the Suns Azimuth from the North at that time. As the Co-fine of the Sun's altitude is to the Sine of the time from Noon (converted into degrees), fo is the Co- line of the Sun's declination to the Sine of his Azimuth from the North, 11. *The Latitude of the place and the Suit s Declination being given, to find the time of the Suns rifing and Jet- ting. Find the afcenfional difference by Problem 4. Then, the degrees of the afcenfional difference being converted into time, fubtradr, that time from 6 hours, when the Sun is in v, s, rt, 55 5 s\ y and n, and the remainder will be the time of Sun-rifing, and added to 6 will be the time of Sun-fetting, But, when [ 77 ] the Sun is in ^ *V* ? and x, the afcenfional difference added to 6 hours gives the time of Sun-rifing, and fub- tra&ed therefrom gives the time of his fetting. How to find his Amplitude at rifing and fetting, is already fliewa by the 5th Problem, 12. The Latitude of the Place and the Suns Declination being given^ to find the Suns Meridian Altitude % Subtract the latitude from 90 de- grees, and the remainder will be the Co- latitude. Then, if the latitude and declination be v both North or both South, the Sum of the declination and Co-latitude is the Sun's Meridian alti- tude. But when either of thefe is North and the other South, their dif- ference is the Meridian altitude. 2* C 78 ] To find the Suns Altitude at any time of the day^ by the Jhadow of an up- right objecl on a horizontal Plane. As the length of the fhadow is to the height of the object, fo is Radius to the Tangent of the Sun's altitude at the time of obfervation, jthe Latitude of the place^ the Suns Me- ridian Altitude^ and prefent Alti- tude^ being given 3 to find the time of the day. As Radius is to the Co-fine of the Sun's declination, fo is the Co-fine of the latitude to a fourth Sine; and, as that fourth Sine is to Radius, fo is the difference between the Sun's meridian altitude and his prefent altitude to the verfed Sine of the time from Noon, The? [ 79 ] The intended Aftronomical Problems being finifhed, we now give the pro- mifed Table of Refradions. Seepag. 72. Appar. Ke frac- Ap. Refrac- Ap. Refrac- *£* Altit. tion. Alt. tion. Ali- Alt. tion. 0 — O 0 r tr 0 0 33 45 2 I 2 18 5 6 O 36 0 15 30 24 22 2 11 57 0 35 0 3° 27 35 2 3 2 5 « Q 5 8 0 34 0 45 2 5 1 1 24 J 59 59 0 32 M 1 0 23 7 2; 54 OO 0 31 1 15 2 1 20 20 1 49 OI 0 30 to 1 30 *9 46 27 1 44 62 0 28 1 45 18 22 28 1 40 6 3 0 27 2 0 l 7 8 29 1 36 64 O 26 2 30 15 2 30 1 3 2 6 S O 25 3 0 '3 20 3 1 1 28 06 O 24 3 3° 1 1 57 3 2 1 2 5 O 23 4 0 10 48 33 1 22 68 O 22 4 30 9 5° 34 1 l 9 69 O 21 5 0 9 2 35 1 16 70 O 20 the rent 5 '30 8 21 3 6 1 13 71 O 19 6 0 7 45 37 1 11 72 O 18 It 6 3° 7 14 3 8 1 8 73 ° 17 7 1 0 30 6 6 47 22 39 40 1 I 6 4 74 75 O 16 0 15 ! 8 0 6 0 4 1 2 76 0 14 <2 8 3 C 5 40 42 0 77 0 13 9 , c 5 22 43 0 58 7» 0 12 9 3 C 5 6 44 0 56 79 O 1 1 lO c 4 5 2 4 £ 0 54 80. 0 10 1 1 c 4 27 46 0 5 2 81 0 9 12 c 4 5 47 0 5° 4 8 82 0 8 •f S •3 c 3 47 48 0 *3 0 7 t-» ^> 4 c 3 31 49 0 47 84 0 6 1 5 c 3 »7 5° 0 45 85 0 5 16 0 3 4 5 1 0 44 86 0 4 17 c 2 53 5 2 0 42 87 0 3 18 c 2 43 53 0 40 88 0 2 '9 c 2 34 54 0 39 89 0 1 20 c 7 26 55 0 38 90 0 0 [8o] *fhe Defcription of an Inflrument for folving many Aflronomical Problems % y fnding the Hour-difta7tces from XII on horizontal and vertical Dials ; formmgfpherical Triangles, and folv- ing the Cafes depending thereon^ &c. Mr Mungo Murray, Shipwright, contrived a very ufeful inflrument fe- veral years ago, which he calls The Ar- millary Trigonometer : and I had it fome months by me in the year 1757. Since that time, he fhewed me a pafte- board model of an inftrument, much of the fame fort, but of a much fmaller fize ; which, I believe, he has not yet made, either of wood or metal. And, as it is a thing that deferves well to be known, on account of its great uti- lity, I have made it of wood, as re- prefented in Plate I. The only addi- tion that I have made to Mr. Murray's fcheme, is a circular fcale of the Sun's declination for the different days of the year, to fave the trouble of referring 7 to [ 8i ] to Tables of the Sun's declination in printed books; as it is one of the data that muft be had in folving moft of the following Problems, which are only a few of thofe that may be folded by it* The upright circular board A is 1 2 inches in diameter, and one inch in thicknefs. It ftands on the horizontal foot J5. On the left hand fide of this board is a flat femicircle C ; which is made of box wood, and is pinned faft to the board A. To this femicircle is joined fuch an- other, Z>, by two hinges, at the zenith and nadir (fo marked in the figure) ; and is moveable on thefe hinges. When D is put down flat to the board* it and the other make a flat circular ring; on which the months and days of the year are laid down, and all the degrees of the Sun's Norh and South declinations anfwering thereto: within which, the four quadrants of the circle are divided into 90 degrees each, M In ■ t 82 3 In the upright board A, the ends of the horizontal femicircle E are fixed. This femicircle ftands at right angles to the plane of the board, parallel to the foot B; and is divided forwards and backwards into 32 equal parts, for the points of the compafs ; within which the two quadrants are divided into 90 degrees each, numbered from the South and North points to the Eaft and Weft, at £. W, where the numbers end at 90. Within the two femicircles C and D (when D is put down) is the flat board i 7 , whofe furface lies even with the furfaces of thefe two femicircles; and which is moveable, round the fixed pin yin the center. On this board is a diameter line (marked Axis) which re- prefents the axis of the world, and ter- minates in the North and South poles; where two hinges join the moveable iemicircle G to the board. This fe- micircle is divided, upwards and down- wards, from the middle to the North and South poles, into twice 90 degrees, for 1 C «3 ] for all the North and South declina- tions of the Sun, Moon, and Stars. In the moveable board jF, the ends of a femicircle H are fixed. The plane of this femicircle is at right angles to the plane of the board, and alfo to the plane of the femicircle G in all po- fitions. It is firft divided into twelve equal parts, where the hours are dou- bly laid down : and then, each hour is fubdivided to every fifth minute. The outermoft hours are thofe from Midnight to Noon, and the innermoft are the hours from Noon to Midnight. A quadrant i, whofe fur face is even with the furface of the great board A, is divided into 90 degrees, numbered upward from the horizon to the zenith. As thefe cemicircles anfwer all the purpofes of whole circles, in the inftru- ment ; we mall call D the vertical cir- cle, E the horizon, G the hour circle, and H the equator. There is a notch in Z), which receives E ; and the in- nermoft edge of D goes clofe to the outermoft edge G, whofe innermoft M 2 edge C «4] edge touches the outer moft edge of H $ in all portions. *£he latitude of the place, and the day of the month being given \ to reBify the Inflrument for ufe. In the following Problems, we {hall always fuppofe the latitude of the given place to be North. Therefore, turn the moveable board F till the North pole comes to the latitude of the place, on the quadrant /; then, find the Sun's declination for the given day of the year, on the femicircles C or D ; and, as that declination is North or South, mark it with chalk, North or South of the equator H y on the move- able hour-circle G; and the inflrument will be rectified: and remember that it mull always be fo, in each of the following Problems, except the 9th 3 and nth. Pros. I, [ 8 s ] Prob, I. To find the time of the day, either in the Forenoon or Afternoon ; and the Suns true Azimuth from the South at that time. Obferve the Sun's altitude with a quadrant. Then, move the vertical circle D> and the hour-circle G y till the Sun's obferved altitude (above the horizon E) on the former, coincides with his declination on the latter: and then, the circle G will cut the time of the day in H when the obfervation was made ; and the number of decrees reckoned from the South point of the horizon E to the vertical circle Z), will be the Sun's true Azimuth from the South at that time. Prob, II, To find the variation of the compafs. The Sun's true azimuth being found hy the foregoing Problem, compare it with C 86 ] with the azimuth fhewn by the compafs at the time of obfervation ; and the difference will be the variation of the compafs at the place where the obfer- vation was made. Prob. III. *The time of the day being given; to find the Suns altitude and azimuth at that time. Put the hour circle G to the given time on H ; and, keeping it there, move the vertical circle D till it cuts the Sun's declination in G; and the interfe&ion will cut the Sun's altitude above the horizon in D, and D will cut the Sun's azimuth from the South in E. Prob. IV. To find the time of the Suns rifing and Jetting) on any day of the year in any given North latitude lefs than 66 \ degrees. The reafon for confining this Pro- blem within 66 \ degrees is, that in greater C 8 7 3 greater latitudes, the Sun continues feveral natural days (of 24 hours each) above the horizon in fummer, without fetting ; and the time is the longer as the place is the nearer to the pole. At the poles of the earth, the Sun is con- tinually above the horizon for the Summer half year, and continually below it for the Winter half. To folve this limited Problem, turn the hour- circle G till the Suns declination there- on, for the given day, comes to the ho- rizon E ; and then, G will cut H in the time of the Sun's rifing, among the outermoft hours ; and the time of his fetting, among the innermoft. Prob. V. To find when the Morning "Twilight be- gins j and when the Evening Twilight ends. When the Sun isjufl 18 degrees be- low the horizon in the Morning, the 2 Twilight [88] Twilight begins; and when he is iS degrees below the horizon in the Even- ing, the Twilight* ends. Therefore, mark the x8 degree below the hori- zon E in the vertical circle Z); and mark the Sun's declination for the gi- ven day in the moveable hour-circle G. This done, turn D and G on their hinges till you find the 18 th degree below the horizon on D cuts the de- clination on G : and then G will cut H in the time when the Morning Twi- light begins, among the ou term oft hours; and the time when the Even- ing Twilight ends, among the inner- moft. IV. B. When the point of the Sun's declination, in the Summer months, does not go fo far as 1 8 degrees be- low the horizon, at Midnight; the Twilight continues all the Night. PROB, VI, C 89 ] PROB. VI. A place being given in the North frigid zone ( that is> in more than bb{ degrees of North latitude) to find on what day of the year the Sun begins to Jhine conflantly on thatplacewithout fetting \ and how long he continues to do jo* The pole being elevated to the la- titude of the place, put down the move- able hour-circle G quite flat to the board A ; and then, obferve what de- gree, or point of North declination on G, cuts the horizon E. When the Sun is at that point of declination, before the 21ft of June, he begins to go on, without fetting ; and continues to do fo till he comes to the like point of de- clination after the 2 1 ft of June. There- fore, the two days, before and after the 2 1 ft of June, which anfwer to the faid point of declination in the fcale of months^ give the folution of the Pro- blem : that before the 21ft of June N being C 9° ] being the day on which the Sun be«ins to go round with fetting ; and he con- tinues to do fo, till the other after the 2 1 ft of June, on which he begins to fet ; and then to rife and fet as at other places. Prob. VII. To find how long the Twilight continues at the poles of the Earth, The continuance is equal at each pole, but at contrary times of the year : fuppofe therefore we take it for the North pole. At the North pole, while the Sun is above the horizon, his altitude is equal to his declination North: and while he is below the horizon, his de- preffion is equal to his declination South ; and his South declination be- gins on the 23d of September, and ends on the 20th of March. But, as his South declination is within 18 degrees from the 23d of September to the 1 3th [ 9i ] of November, the Twilight continues all that time after the Sun fets. And as it is within 18 degrees from the 29th of January to the 20th of March, there is Twilight all that time at the North pole before the Sun rifes to it. Prob. VIII. To find the Suns deprejfion below the ho- rizon, at any time of the night, in any given latitude lefs than bb\ degrees. The inftrument being rectified, bring the moveable circle G to the given time of the night in H; then, move the vertical circle D till it cuts the Sun's declination for the given day in G, and the declination in G will cut the num- ber of degrees of the Sun's depreffion below the horizon at that time, in D. N 2 Prob. IX.. # [ 92 ] Prob. IX. To find in what North latitude the long- ejl day is of any given length lefsthan 24 hours. In Northern latitudes, the longeft day is when the Sun's declination is 2 3 \ degrees North. Divide the given length by two, and the quotient will give the time of Sun-fetting: to which time, place the circle G among the in- nermoft hours on H\ and then, turn the moveable board F till 23 \ degrees of North declination on G comes down to the horizon E\ and the elevation, of the pole above the horizon will (hew the latitude of the place, in the qua- drant /. Prob, X. To find the Suns amplitude at rifing and Jetting, in any given latitude lefs tha?i 66\ degrees. The time of the Sun's rifing and fet- ting being found by Prob, IV^tount the number [ 93 ] number of degrees on the horizon E y at which he rifes and fets from the Eaft and Weft points; and t hat number will be the Sun's amplitude. Prob. XL The length of the longefl day being given % at any place whoje latitude is North ; to find the latitude of that place. If the given length be lefs than 24. hours, fubtract its half from 12 hours; and the remainder will be the time of Sun fetting on that day. To which time, place the moveable hour-circle G y among the innermoft hours on H; and then, turn the moveable board F till 2^ d egrees of North declination, on G comes down to the horizon E y and the pole will then point out the lati- tude of the place, in the quadrant /: the elevation of the pole above the ho- rizon of any place being always equal to the latitude of that place. If [ 94 ] If the length of the longeft day con- fills of feveral natural days, of 24. hours each ; take the Sun's altitude by a qua- drant when he is due North, on the 21ft of June; at which time his de- clination is 23^ degrees North. Then, put down the femicircles D and G, flat to the board A ; and turn the move- able board jP till 23^ degrees of North declination on G cuts the Sun's ob- ferved altitude on D : and then, the North pole will point to the latitude of the place, in the quadrant % Prob. XII. In the Summer months ^ to find ct7i Eajl a?id Weft line, and confequently a Meridian Itns^ for a place of any gi- ven latitude. This is beft done about the time of the Summer folftice ; becaufe the Sun's decimation changes flower about that time than any other in Summer : and it cannot be done in the Winter-half of [ 95 ] of the year, by the inftrument; be- caufe, during that time, the Sun is al- ways paft the Eaft before he rifes ; and he fets before he comes to the Weft. Having fet a wire upright in a level board, on which the Sun may fhine when he is due Eaft or due Weft, as already mentioned (pag. 70) and the inftrument being rectified, bring the vertical circle D to the Eaft and Weft point of the horizon £, and turn the moveable hour-circle G till the Sun's declination thereon, for the given day, comes to the vertical circle D ; and the Decli- nation in G will cut the Sun's altitude in D when he is due Eaft or due Weft on that day. For the reft of the ope- ration, fee pag, 70 and 71. Prob. XIN. 7# find the dijlances of all the Forenoon and Afternoon hours from XII, on a horizontal dial for any given Lati- tude. Elevate the pole in the quadrant 7, to the latitude of the given place ; and 3 bring C 96 ] bring the moveable hour-circle G fuc* ceffively to all the outermoft hours on the equator H\ and G will cut the diftances of all the Forenoon hours from XII on the horizon E, as you bring it to the like hours on the equator. The Afternoon hours being at the lame diftances from XII as the Forenoon hours are, by having the latter we have alfo the former, N. B. When 23- degrees of North declination on G comes to the ho- rizon E y G will cut H in the time of the Sun's rifing and fetting on the longeft day ; and confequently will limit the number of hours to be put upon the dial. By the fame method, the half hour and quarter diftances from XII may be found, for all the hours on the dial. Paob, XIV. [ 97 ] Prob. XIV. Tifind the diftances of the Forenoon and Afternoon hours from Xll, on a ver- tical South dial for any given lati- tude. The pole being elevated in the qua- drant /, to the latitude of the given place, bring the vertical circle D to the Eaft and Weft point of the horizon E (at E. W.) This done, bring the move- able circle G fucceffively to all the out- ermoft hours on H y from XII to VI ; and G will cut Din all thefe hour-di- ftances from XII, reckoned downward from the zenith to the horizon. Thefe are the Forenoon hours: and as all the Afternoon hour-diftances from XII to VI are the fame as the Forenoon hour-diftances, 'tis needlefs to work for them by the inftrumenr. N. B. On all erecl dired South dials, the Forenoon hours begin at VI in the Morning ; and the Afternoon O hours C 93 ] hours end at VI in the Evening : for the Sun never mines more than twelve hours on any dial whofe plane is perpendicular to the horizon. The meridian, or twelve o'clock line, on thefe two dials, muft be made as broad as the ftile is thick. The angle of the ftile's height muft, be equal to the latitude of the place for which the horizontal dial is made ; and the angle of the ftile's height in an erect direct South dial muft be equal to the Co-la- titude of the place. Having thus found the hour-diftan- ces from XII by the inftrument, fet them off with your compaffes, by a line of chords, from the twelve o'clock line on your dial plate ; which line being made as broad as the ftile is thick, fet off the Forenoon hours from the edge of the twelve o'clock line, which is to the Forenoon ftde of the dial ; and the Afternoon hours from the Afternoon edge of that line. The [ 99 ] The fix o'clock line is perpendicular to the meridian line on thefe dials. It muft be drawn before you begin to let off the hour-diftances on the dial : and the centers of the two quad ran tal arcs (taken equal to the chord of 60 degrees on your fbale) muft be in thofe points of the edges of the broad me- ridian line, where the fix o'clock line interfects it. And the broad edge of the ftile, that (hews the time by the fhadow, muft rife from thofe points in the dial which were made the centers of the above quadrantal arcs whereon the hour-diftances are fet off from XIL Pros. XV. To find the dijlances of the Forenoon and Afternoon hours from XII, on a ver- tical dial, declining from the South toward the Eajl or Weft, by any gi- ven number of degrees. Let us fuppofe that face of the dial muft decline (or turn away) 30 degrees O 2 from C 100 ] from the South toward the Eaft ; then, 'tis evident, that the Afternoon edge of the plane of the dial is 30 degrees from the Eaft toward the North, and the Forenoon edge thereof is 30 de- grees from the Weft toward the South. Therefore, count the 30 degrees of declination from the Eaft point of the horizon E (at 90, under E. W.) to- ward the North point - $ and where the reckoning ends {viz. at 60 degrees from the North), place the vertical circle & in the horizon ; and D will reprefent the plane of the declining dial. Then, to find the diftances of the Forenoon hours from XII on the dial, bring the moveable hour-circle G fuccefiively to all the Forenoon hours (which are the ou term oil) on the equator //, as XI, X, IX, &c. till G comes to the horizon E in the point where the vertical circle D interfecls it. And in doing this, G will cut the diftances on D of all the Forenoon hours from XII, that muft be put upon the dial j thefe diftances being [ "0# 1 being reckoned downward on Z), from the zenith to the horizon. Then, becaufe we hav T e only femi- circles in the inftrumenr, to find all the hour-diftances by ; and as the Afternoon hours are not equidiftant with the Fore- noon hours from XII on declining di- als; let the Afternoon hours I, II, III, &c. be reckoned among the outer mo ft on the equator H from the right hand toward the left, and let their diftances from XII be taken upward on the ver- tical circle D, from the nadir toward the horizon E. To find thefe diftan- ces, bring the moveable hour-circle G fuccemVely to the hours I, II, III, &>c> (which are outermoftj on the equator, till it comes to that point of the ho- rizon E where the vertical circle D interfects it ; and G will cut D in the diftances (reckoned upward from the nadir) of all the Afternoon hours that muft be inferted on the dial. Having thus found the diftances from XII, of all the Forenoon and Af- ternoon hours that muft be inferted 1 on t 102 ] on the declining dial, and wrote them down ; draw a fingle line (no thicker than the other hour lines) acrofs the plane of the dial, perpendicular to that edge of it which muft be the lowermoft, and be parallel to the horizon, when the dial is fet; and that line, perpen- dicular to the horizon, will be the Me- ridian or twelve o'clock line of the dial. Near the uppermoft end of that line, affume a point for the center of the dial; and, having taken 60 degrees from the line of chords in your com- paffes, fet one foot in the center-point, and with the other foot defcribe a fe- micircle on the dial-plane ; and there- on fet off all the above-found diftances from the XII o'clock line ; and place the hours at thefe diftances accordingly. Then, to find the diftance of the fubftile (or line on which the ftile muft ftand) from the meridian line of the dial; bring the moveable hour-circle G to as many degrees from the South toward the Eaft point of the horizon E as the vertical circle D ftand s at from [ I0 3 ] from the Eaft toward the North point ; and the circles D and G will crofs each other at right angles. Then, the num- ber of degrees on Z>, which are inter- cepted between G and the zenith, will be the angle that the fubftile makes with the meridian of the dial ; which muft be fet off from XII, among the Forenoon hours, -becaufe the face of the dial declines from the South toward the Eaft. And the number of degrees on G, which are intercepted between the vertical circle D and the North pole, will be the angle of the ftile's height. The fubftile line muft be drawn to the center point of the dial, that is, to the center of the femicircle on which the hour-diftances were fet off from XII; and the edge of the ftile that fhews the hours by the fhadow, muft begin to rife from the dial at the center point. In this dial, the ftile muft be very- thin, or elfe have a (harp edge. If the dial declines Weftward from the South, the vertical circle D muft be placed C I0 * ] placed as many degrees from the Eaffc point of the horizon, toward the South, as the dial declines : and then, the hour-diftanccs from XII are to be found in the fame manner as above defcribed. In E aft- declining dials, the fubftile falls among the Forenoon hours ; and in Weft- declining dials, among the Af- ternoon hours. For, in all kinds of dials, when they are properly fet, the edge of the ftile, that cafts the fhadow for mewing the time of the day, mud be parallel to the earth's axis. Every one who reads this, and un- derftands the ufeof the globes, will eafily fee that thefe are only a few of the Problems which may be folved by this inftrument. And a bare view of the figure of it is fufficient to mew, that any fpherical triangle may be readily formed and folved by it ; and confe- quently, all the Problems that depend on fpherical trigonometry. That juftice might have been done to it, I wifli Mr. Murray himfelf had defcribed it, and {hewn all its ufes. To knoWy by the Stars, whether a clock goes true or not. The ftars make 366 revolutions from the Me- ridian to the Meridian again, or from any point of the compafs to the fame point again, in 365 days ; and therefore they gain a 365th part of a revolution every 24 hours of mean folar time. Confequently, if you mark the precife moment fhewn by a clock, when any ftar vanifhes behind a chimney, or other ob- ject, as feen through a fmall hole in a thin plate of metal, fixed in a win- dow-fhutter ; and do this for feveral nights together (as fuppofe 20) if at the end of that time the ftar vanifhes P as A Tab!< : fliewirg" the daily accelera- tions of the Stars. Accelerations. Days. ? a 1 O 3 55 54 2 O 7 51 48 3 O 11 47 42 4 0 J 5 43 3 6 5 0 l 9 39 3° 6 O 23 35 24 7 O 27 31 iS 8 O 31 27 12 9 0 3 5 2 3 6 10 O 39 ! 9 0 1 1 O 43 H 54 1 2 O 47 10 48 13 O 51 6 42 *4 O 55 2 3^ 1 5 O 58 58 30 16 I 2 54 24 »7 18 I 6 50 18 10 46 12 l 9 14 42 6 20 18 38 0 21 J 22 33 54 22 26 29 48 2 3 30 25 42 H 34 21 3 6 25 38 17 30 26 42 13 24 2 7 46 9 18 28 50 5 12 29 54 1 6' 57 57 0 [ io6 ] as much fooner than it did the firft pight, by the clock, as anfwers to the accelerations for fo many days in the Table ; the clock goes true ; otherwife not. If the difference between the clock and ftar be lefs than the Table fhews, the clock goes too faft ; if great- er, it goes too flow; and muft be re- gulated accordingly, by letting down or raifing up the ball of the pendulum, by little and little, till you find it keep to true equal time. Thus, fuppofe the ftar fhould dif- appear behind the chimney any night when it is XII by the clock; and that, pn the 20th night afterward, the fame ftar mould djfappear when it is 41 mi- nutes, 22 feconds, paft X by the clock, which fubtracled from XII h. o m. o f. leaves remaining 1 hour, 18 minutes, 38 feconds, for the time the ftar is. then fafter than the clock; look in the Table, and againft 20, in the left- hand column, you will find the acce- leratipn of the ftar to be 1 hour, 18 min. 38 feconds; agreeing exactly with. what [ *°7 ] what the difference between the clock and ftar ought to be; which £hews that the clock meafures true equal time. Dr. Defagulkrs informs us, that the length of a pendulurn (from the point of fufpenfion to the center of ofcillation) that fwings feconds in the latitude of London, is 29.128 inches. Now, to find the length of a pendulum that fhall make any other given number of vibrations in the fame latitude, in a minute ; fay, as the fquare of the given number of vibrations is to the fquare of 60, fo is 39.128 inches (the length of the ftandard) to the length in inches of the pendulum fought. By th is rule, the following Table is calculated, for all numbers of vibrations in a minute, from 1 to 180, fervins for the latitude of London. Arid, by the next Table that follows (page 1 1 o) the peridulum may be corrected for any other latitude. P 2 A Tables C *°* ] A Table, Jhewing of what length a Pendulum mujl be, to make any given number of Vibrations in a minute, from i to i8o, in Lat. 5 I<5 3°'- Length of the Pendulum. Feet. Inches. i I 2 3 4 5 6 7 8 9 10 1 1 I 2 !3 1 4 1 • 16 i? 1 8 19 20 2 i 22 23 *4 25 2 1 2 7 zS *9 1738 2 934 I3°4 73.3 4 6 9 326 239 183 U4 1 16 97 80 69 59 52 45 40 36 32 29 26 24 22 20 18 17 16 »4 13 1 1 4.800 7.200 3.200 7.800 6.432 0.800 6.710 4.800 1 1.022 8.608 0.139 11.255 2.538 5.566 2.048 10.234 7.408 2- 755 6.196 4.154 7- 6 39 3- °34 . 2.277 4.550 8.377 4- 174 1.238 1 1.669 1 1 .492 > 0.512 ^ Length of the Pend. 3 1 3 2 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 ;o 5' 5 2 53 54 55 56 57 58 59 60 Feet.Inches. 2.577 5- 6 37 9-358 1.852 6.998 o;68i 6.894 1-543 8.61 1 4.038 1 1.736 7- 753 4.074 ' °-759 9.561 6.569 3^-767 »-i37 10.607 8- 344 6.156 4-093 2.146 0.169 10.465 8.818 7-355 5.873 4.466 3.128 61 6 2 63 64 65 66 67 68 69 70 7 1 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Length of the Pend. Feet.Inches 1.852 0.644 1 1.490 10.387 9.340 8.337 7.38i 6.465 5.586 4«747 3-943 3.172 2-433 1.723 1.042 0.387 11.758 11.152 10.571 10.009 9.469 8.949 8.447 7-9 6 3 7.496 7.045 6.610 6.190 5-771 5 390 C I0 9 ] The Pendulum Table concluded. < Length of 7 112 0 1 1.2 I I 142 6.985 172 4.761 113 0 1 1 .03 I 143 6.888 •73 4.707 114 0 1 D.8 3 8 144 6 -793 »74 4.652 Ho- 0 iQ.561 145 6.699 *75 4-599 ne 0 10.368 146 6.608 176 4-547 117 0 10.281 *47 6.518 l 77 4496 118 0 1 0.1 17 148 6.43 1 178 4-44 5 u s 0 9.954 149 5*345 179 I2C » 0 9.782 *5° 6.260 180 1 4- 347 Length for 300 Vibrations in a min. 1 inch and .565 parts of an inch. s - B- 3 n 1 £ CTQ ' ST o o ^ 2T cl w « c 75 cr p* CO c/q' cr p © a 4 o> o I 7 (1 cr cr 3 co >-! o> o 2.0 OQ f* ^ 2" rt CT) fcr P Cl n CL c 3 cr rt o Si A Table, jhewing how much a Pendu- lum that fwings Seconds at the Equator would gain every 24 hours in different Latitudes ; and how much the Pendulum would need to he lengthened in theje Latitudes, in order to make it fwing feconds there- in. Latitude of the Place. Deg. 5 10 15 20 25 3° 35 40 A pendulum that fwings feconds at the equator muft be ~~ parts of an inch fhorter than one that fwings feconds at London ; and a pendulum that fwings feconds at the poles muft be ~ parts of an inch longer than one that fwings feconds at London. The caufe of this difference arifes from the fphe- roidical ■ Tim< Lengthen- gained ing of the in one Pendulum Day. to fwing Seconds. iSeconds. Inch. Parts. 1 •7 0 .0016 6 •9 0 .0062 . 15 •3 0 .0138 26 •7 0 .0246 . 40 .8 0 .0369 j 57 .1 0 .0516 i 75 .1 0 .0679 94 •3 0 .0853 IH4- .1 0 .1033 Latitude of the Place. Deg. 5° 55 60 65 70 75 80 85 90 Time gained in one Day. Seconds. Lengthen- ing of the Pendulum to fwing Seconds. Iach. Parts. 134 '53 171 187 201 2 *3 22 1 226 228 .0 .2 .2 •5 .6 .0 •4 •5 •3 .1212 .1386 .1549 .1696 .1824 .1927 •2033 .2050 .2065 [ III ] radical figure of the earth, and the centrifugal force diminishing (and fo ading gradually, lefs and lefs), from jthe equator to the North and South poles, as the diurnal motions of the places are flower and flower. The length of a pendulum that fwings feconds at the equator is 39 inches ; and the length of a pendulum that fwings feconds at the poles, is 39.266 inches. /I Defcription of three uncommon hinds of Clocks. I. About twenty years ago, I made a wooden model of a clock, which fliews the apparent diurnal motions of the Sun, Moon, and Stars, with the times of their rifing, fouthing, and ■ fetting, for every day of the year; to- gether with all the various phafes of the Moon, and times of her being New and Full in the different months of the year; with the days of the months, never needing to be fhifted, fave once 2 in C ] in four years ; and the age of the Moon for every day of the year, and to every third hour from her change, in any current Lunation. All thefe are fhewn on the dial-plate, without any confufi- on ; and I keep the model ftill by me to fhew in my lectures. The outer part of the dial- plate is divided into twice twelve hours, and each hour into eight equal parts, for the half hours, quarters, and half quarters. Within this circle of hours there is a ring* which goes round once in 24 hours, and carries an index for pointing to the hours of the day and night, and a gilt ball for reprefenting the Sun, and (hewing his apparent diurnal motion round the earth. This ring is divided into 29 days, 1 2 hours, and 45 minutes, for the Moon's age from change to change. A ball, half black, half white, is turned round its axis in 29 days, 12 hours, and 45 minutes, for (hewing all the various phafes of the Moon : the axis of the ball lies in the plane of $he ring, and comes out a little way t *U ] beyond the Moon, and points to ne? age in the forefaid divifions on the ring, falling back every day as much as the Moon is later of coming to the meridian every day than ftie was on the day be- fore; and confequently, the Moon falls back every day fo far in the ring, as td go round it in 29 days 12 hours 45 minutes, from the Sun to the Suri again i Within this ring is a flat circular plate, divided all around its edge into 365 equal parts, for the months and days of the year, which are fet at the proper divifions. This plate makes 366 revolutions (as the ftars do) in the time the Sun makes 365; by which means, the wire that carries the Sun round in 24 hours, cuts the day of the month on the plate, as the plate ad- vances a 365th part of a revolution upon the Sun, once every 24 hours : fo that the plate turns round in a fy- dereal day, which confifts of 23 hours* 56 m, 4 f. 6 thirds, of mean folar time, and [ "4 3 and the Sua goes round in 24 mean folar hours. The equator, ecliptic, and tropics are drawn on this plate; the ecliptic is divided into the 12 figns of the Zo- diac, and each fign into 30 degrees. The wires, which carry the Sun and Moon, cut their places in the Ecliptic, for every day of the year. All the re- markable ftars of the firft and fecond magnitudes are laid down on this plate, according to their right afcenfions and declinations. Over this plate is a fixt horizon for fhewing the times of the rifing and fetting of the Sun, Moon, and Stars. When any ftar comes to the Eaft fide of the horizon, it rifes ; and the hour- hand points to the time of its rifing : when it fets on the Weftern fide of the horizon, the hour-hand points out the time of its fetting ; on any day of the year which the Sun's wire then cuts. When the points of the ecliptic, which are cut by the Sun's wire, and the Moon's, come to the Eaft and Weft fides [ "5 ] fides of the horizon, the hour-hand points to the times of their rifing and letting, on the day of the year which the wire then cuts that carries the Sun. The wheel- work of this clock is as follows : In the center, behind the middle of the Dial-plate, is a fixed pinion of 1 6 leaves, round which one wheel of 100 teeth, and another of 70, are carried, every 24 Ihours; the leaves of the pi- nion taking into the teeth of the wheels. On the axis of the wheel of 100 teeth is a pinion of 14 leaves, which turns a wheel of 69 teeth, on whofe axis is a pinion of 7 leaves, turning a wheel of 83 teeth, which wheel is pinned to the back of the fydereal flat plate above-mentioned, which has the months and figns, &c. upon it ; and any given point in the edge of this laft wheel and plate, revolves from the meridian to the meridian again, in 23 hours, 5 6 min. 4 fee. 6 thirds (which makes a fydereal day) and from the Sun to the Sun again (which revolves in 24 hours), Q^z in in 365 days, 5 hours, 48 minutes, 58 feconds, and 47 thirds, of mean folar tirrie. The above-mentioned wheel of 70 teeth (which is carried round the fixt pinion of 16 leaves every 24 hours) has a pinion of 8 leaves on its axis, which turns a wheel of 54 teeth ; and to the axis of this wheel of 54 the Moon's wire is fixed, which carries the Moon round, from the meridian to the meridian again, in 24 hours*, 50 min. 2 1 feconds ; round the ecliptic on the flat plate in 27 days, 7 hours, 43 mi- nutes, and round from the Sun to the Sun again (or from change to change} in 29 days, 12 hours, 45 minutes. II. About ten years ago, I made a wooden model of a clock for fhewing the apparent diurnal motions of the ■* It is generally believed that the Moon re- volves from the meridian to the meridian again, in 24 h. 48 min. but that is a miftake : for if me did, there would be 30 complete days from change to change. Sun [ **7 ] Sun and Stars, with the times of their rifing and fetting for every day of the year ; and the days of the months all the year round, without any need of fhifting by hand in the fhort months, as is always done in common clocks. I copied the Dial-plate of this model from a clock that Mr. Ellicott had made for the king of Spain : but al- though Mr. Ellicott fhewed me the whole infide of the clock, I did not ailc him what the numbers of teeth in the wheels of it were, although, I am convinced, he would have told me, if I had ; nor do I, in the leaft, remember how many wheels there were in the uncommon or Aftronomical part of it; and fo I fet about contriving wheels and numbers for performing the like motions. The Dial-plate contains twice twelve hours, and within the circle of hours there is a large opening in the plate, a little elliptical : the edge of this open- ing ferves for an horizon. Below the Dial-plate, and feen through the large opening in it, is a fiat [ n8 ] flat plate on which the equator, eclip- tic, and tropics, are drawn ; and all the ftars of the firft, fecond, and third magnitudes are laid down, that are vi- lible in the horizon (of Madrid in Mr. Ellicott's, and of London in mine) ac- cording to their right afcenfions and de- clinations : the center of the plate being the North pole. The ecliptic is cut out into a narrow groove in the plate ; and a fmall Sun Aides in the groove by a pin, and is carried round by a wire flxt in the axis, which comes a little way out through the center of the plate. The edge of this plate is divided into the months and days of the year, and the Sun's wire fhews the days of the months in thefe divifions. This ftar plate goes round in a fydereal day, making 366 revolutions in a year; in which time the Sun makes 365, and confequently fhifts a divifion, or day of the month, every 24 hours. A fmall wire is flretched from over the center of the fydereal plate to the tipper [ iig ] upper XII on the fixed Dial-plate, This wire is for the meridian. When the Sun, or any ftar, comes to the Eaftern edge of the horizon, the hour index is at the time of rifing of the Sun or Star, for the day of the year, pointed to by the wire, that car- ries the Sun : and when the Sun or Star comes to the Weftern edge of the horizon, the hour index is at the time of its fetting. The Sun always comes to the meridian at the inftant of the folar noon ; but every ftar comes fooner to the meridian every day, than it did on the day before, by 3 minutes, 5 5 feconds, 54 thirds of mean folar time, as it revolves from the meridian to the meridian again in 23 hours, 56 m, 4 f . 6 thirds. When any ftar is on the meridian in the clock, the ftar which it reprefents is on the meridian in the heavens ; the time whereof is feen by the hour index on the Dial -plate. And, as* the ftars have their revolutions on the plate, one may look at the clock at any time, and fee t 120 ] fee what ftars are then above the hori- zon, what ftars are then on the meridian^ and what ftars are then rifing andfettingo My contrivance for fhewing thefe motions and phenomena, in the model, confifts of no more than two wheels and two pinions, as follows : The wheels are of equal diameters, and fo are the pinions ; the numbers of teeth are 61 in one wheel, and 73 in the other. The pinions are both fixt on one axis, the one having 20 leaves and the other 24. The wheel of 73 teeth is fixed to the back of the fydereal wheel, and the axis of the wheel' of 61 comes through t the wheel of 73, and through the fydereal plate, and carries the wire round on which the Sun fides in the ecliptic groove, and alfo the hour hand on the Dial-plate. The wheel of 61 teeth turns the pi- nion of 20, and the pinion of 24, (fixt on tjie fame axis with that of 20} turns the wheel of 73. Now, if the wheel of 61 teeth be turned round in 24 hours, to carry the 2 Sun f 121 J Sun and hour hand, the wheel of 73 teeth will be turned round in 23 hours, 56 minutes, 4 feconds, 6 thirds. And lo, the fydereal plate will make juft 366 revolutions, in the time that the Sun makes 365. Mr. Ellicott had the prettieft, and moft fimple contrivance I ever faw, in his clock, for fhewing the difference between equal and folar time (generally- termed the equation of time) on all the different days of the year. He gene- roufly allowed me to copy that part into my model, and I have quite con- cealed it within one of my wheels, not to (hew how it is done unlefs he pub- limes an account of it. The Sun, by that fimple contrivance, even in my model, comes as much fooner or later to the meridian, than when it is Noon by a well-regulated clock, as the Sun in the heavens does, at all the different times of the year, excepting the four days on which the time of Noon fhewn by the Sun and clock ought to coin- cide : and then there is no difference in R the C i« ] the clock. And although the wheel- work is quite open to fight in the model which I now fhew in my lec- tures, no perfon who fees it can guefs how the unequal motion of the Sun, in the model, is performed. III. In the year 1764, when I happened to he at Liverpool, I con^ trived a clock for Captain Hutchinfon, who is Dock- matter of the place, for (hewing the age and phafes of the Moon, and the time of High and Low water at Liverpool, every day of the year, with the ftate of the tides at any time of the day; by looking at the clock. At the right and left lower corners of the Dial-plate, under the common circles for the hours and minutes, there are two fmall circular plates. On the plate at the left hand there are two circular fpaces, the outermoft of which is divided into twice twelve hours, with their halves and quarters : within which, the lecond circular fpace is divided into 19^ equal parts for the days of the Moon's, [ I2 3 ] Moon's age; each day (landing under the time of the Moon's coming to the meridian on that day, in the circle of 24 hours* An axis comes through the center of this plate, and carries two in- dexes round it in 29 days, 12 h. 45 min. or from change to change of the moon : and thefe indexes are fet as far afunder, as the time of High water at Liverpool differs from the time of the Moon's corning to the meridian. So that, by looking on this plate in the Morning, one may fee at what time the Moon will be on the meridian, and at what time it will be High water at the place. On the right hand plate, around its edge, all the different ftates of the tides are marked, from High to Low, and from Low to High ; and within thefe appellations is a (haded ellipfis, the higheffc points of which reprefent High water, and the lowed parts Low water. An index goes round this plate in the time of the Moon's revolving from the meridian to the meridian again ; and, R 2 at [ ] at all different times, points out the ftate of the tide, as it may be then High or Low, rifing or falling. In the arch of the Dial-plate above the hour of XII, a blue plate rifes and falls as the Tides do at Liverpool : and, over this plate, in a painted fky, a glo- bular ball, half black, half white, (hews the phafes of the Moon for every day of her age, throughout the year. The wheel-work for fhewing thefe appearances is as follows. A wheel of 30 teeth is fixed on the axis of the twelve-hour hand, and turns round with it. This wheel turns a wheel of 60 teeth round in 24 hours, and on its axis is a wheel of 57 teeth* which turns round in the fame time, and turns a wheel of 59 teeth round in 24 hours 50 ^ minutes, on whofe axis is the index on the right hand corner-plate, going round the plate in the time of the Moon's revolving from the meridian to the meridian again; and fhewing the ftate of the Tide at any time, when the clock is looked [ »s ] looked at. On the axis of the fame wheel of 59 teeth is fixed an elliptical plate, which raifes and lets down the Tide-plate in the arch, twice in 24 hours 50^ minutes ; in which time there are two ebbings and flo wings of the Tides. The above-mentioned wheel of 57 teeth has a pinion of 16 leaves on its axis, turning a wheel of 70 teeth, on whofe axis is a pinion of 8 leaves, turn- ing a wheel of 40 teeth, which turns a wheel of 54 teeth round in 29 days, 1 2 hours, 45 minutes ; and on the axis of the wheel of 54 teeth are the two indexes on the left hand corner- plate, for mewing the Moon's age on that plate, with the time of her fouth- ing, and of High water. The wheel of 40 teeth here mention- ed, might have been of any other num- ber, and might have been left out, if the pinion of 8 leaves had taken into the wheel of 54 teeth : but then the index would have gone the wrong way round the dial-plate. So that the only [ »6 ] tife of the wheel of 40 is to be a lead- ing wheel, for turning the index round the right way. On the axis of the wheel of 54 teeth (which turns round in a lunation) is a fmall wheel of 20, turning a contrate wheel of the fame number, on whofe axis is the globular Moon (half black, half white) in the arch, turning round in a lunation, and fhewing all her phafes. In thefe three clocks, I have only defcribed the uncommon parts, which are connected with the common part of the movement known unto every Clock- maker. An eafy way of reprefenting the apparent diurnal motions of the Sun and Moon in a clock. Let a thick wheel of 5 7 teeth be turned round in 24 hours, and take into the teeth of two wheels of equal diameters, one of which has 57 teeth and the other 59 ; thefe wheels lying 5 clofe i C l2 7 ] clofe upon one another, and the axis of the one turning within the axis of the other. A wire fixt on the axis of the wheel of 5 7 will carry a Sun round in 2 4 hours ; and a wire fixt on the axis of the wheel of 59 will carry a Moon round in 24 hours 50^ minutes. If the Sun carries a plate round with him in 24 hours, and the limb of the plate be divided into 29! equal parts for the days of the Moon's age, the Moon will fhew her age in the divi- fions of that plate; and may be made to turn round her axis, and fhew her phafes, by a wheel of any number of teeth, on her axis, and taking into the teeth of a contrate wheel of the fame number, fixt on the axis of the wheel of 57 teeth, which carries the Sun. An eafy way of /hewing the phafes of the Moon y in a clock. Let a wheel of 16 teeth be fixed on the axis of a wheel of 1 5, and the, wheel qf 16 turn a wheel of 63, on whofe [ X28 J whofe axis let a ball, half black, half white, be fixed; and project half way- out, through a round hole in the dial- plate. Then, if the wheel of 1 5 teeth be always moved one tooth in 1 2 hours, the ball will be turned round in 29 days, 12 hours, 45 minutes, and (hew all the various phafes of the Moon, An eafy method of Jbewing the Suns place in the Ecliptic every day of the year, in a clock \ and his motion round the Ecliptic in a Solar year. Let a pinion of 12 leaves be turned round once every ten hours, and this pinion take into a wheel of 67 teeth, on whofe axis let there be a fingle threaded fcrew taking into a wheel of 157 teeth. This laft wheel will turn round in 365 days, 5 h. 49 m. 50 fee. And an index on its axis will carry a Sun through the whole 360 degrees of an ecliptic, engraven on the dial- plate, in the fame time : and may fhew [ ] the days of the months on another cir- cle within the ecliptic. This was the contrivance of Mr. Arnfhaw near Man- cbejler, who communicated it to me. How to Jhew the periodical revolutions of the Earth, and all the other planet s y round the Sun> in a Clock ; fo as to agree nearly with the periodical revo- lutions of the planets about the Sun in the Heavens \ Let fix hollow foeketsj or arbors* be made to fit and turn within one ano- ther, and all of them to turn upon a fixt fpindle, or axis ; on the top of which let there be a ball to reprefent the Sun. Let the wideft arbor be the ftiorteft, and have an arm on its up- permoft end to carry a ball reprefenting Saturn, and a wheel of 206 teeth on its lowermoft end. Let the next fized arbor be fo much longer than the above one, as to have a wheel (of 83 teeth) put upon it, be- low the wheel of 206 j and an arm on S tha [ *3° ] the other end (above Saturn's) for car- rying a ball to reprefent Jupiter. Let the third focket be fo much longer than the fecond, as to have a wheel on it (of 47 teeth) below the wheel of 206, and an arm on its other end, above Jupiter's, for carrying a ball to reprefent Mars. Let the fourth arbor be fo much longer than the third, as to have a wheel (of 40 teeth) on its lower end, and an arm on its upper end, above Mars's, for carrying a ball to reprefent the Earth. Let the fifth arbor be fo much longer than the fourth, as to have a wheel (of 32 teeth) on its lower end, below the wheel of 40, and an arm on its upper end, above the Earth's, for carrying a ball to reprefent the planet Venus. Let the fixth (which is the fmalleft) arbor, be fo much longer than the fifth, as to have a wheel (of 20 teeth) on its lower end, below the wheel of 32, and an arm on its upper end, above Venus's, [ 13" ] Venus's, for carrying a ball to reprefent the planet Mercury. Saturn's arm muft be the longed of all, becaufe that planet is the furtheft of all from the Sun : Jupiter's the next longeft, Mars's the next, the Earth's the next, Venus's the next, and Mercury's the next, or ftiorteft of all, becaufe Mercury is the neareft of all the pla- nets to the Sun. The wheels muft be fixed on their refpec~tive arbors, and diminifh in their fizes from the higheft numbers to the loweft; fo that, when they are all put together, they may form fomewhat of the appearance of a cone. And, to give thefe wheels and pla- nets their proper motions, they muft be turned by fix wheels (or rather four wheels and two pinions) all fixed on one folid axis, in a conical manner, inverted with refpect to the other fix wheels; fo as the wheels and pinions on the folid axis may take into thofe on the arbors, and turn them. S 2 The [ 132 ] The folid axis, with all its wheels and pinions, will turn round in the fame time together, becaufe the wheels and pinions are all fixed on the axis 5 and mull be turned round once in a year by the clock-work ; which may be eafily done by fuch a method as Mr. jdrnjhaw s, already mentioned. Then, if the uppermoft and fmalleft pinion on the axis has 7 leaves, taking into Saturn's wheel of 206 teeth ; Sa- turn will be carried round the Sun in 1074.8 days, 18 hours, 43 minutes: for, as 7 is to 206, fo is 365.25 to 10748.78. If the next pinion on the axis (which mu ft be of a bigger fize than the pinion of 7 above it) has 7 leaves, and takes into Jupiter's wheel of 83 teeth ; Ju- piter will be carried round the Sun in 4330 days, 19 hours, 40 minutes: for, as 7 is to 83, fo is 365.25 to 4330.82. If the wheel below this pinion or* the axis has 25 teeth, and takes into Mars's wheel of 47 ; Mars will be car- ried round the Sun in 686 days, 16 hours^ C *33 I hours, 5 minutes : for, as 25 Is to 47, fo is 365.25 to 686.67. If the next bigger wheel on the axis, which turns round in 365.25 days, has 40 teeth, apd takes into the Earth's wheel of 40 teeth; the Earth will be carried round the Sun in 365.25 days. If the next bigger wheel has 5 2 teeth, and takes into Venus's wheel of 3 2 teeth; Venus will be carried round the Sun in 2 24 days, 1 8 hours, 29 minutes : for, as 52 is to 32, fo is 365.25 to 224.77. And laftly, if the largeft wheel on the axis has 83 teeth, and takes into Mercury's wheel of 20; Mercury will be carried round the Sun in 88 days, 6 hours, 14 minutes ; for, as 83 is to 20, fo is 365,25 to 88.01. I have feen a calculation of this fort in a printed book ; but the numbers there are fo faulty for Mars and Saturn, that I was obliged to alter them ; Sa- turn's period being wrong by 5 1 days. How near thefe are to the truth will ap- pear by comparing them with the an-* nual periods in the following Table. ATabk [ r 34 ] oj oc On M 00 „ SO __ -»» u v • o ^ w v © m o b» oo a O O O ~ «-! S" S"H. 5 » S I* 2. **: 5^ H Er s* =r 3 3 « fD W y ? | 5 » a. >r ? ?■ a a- ?r 3 ' '3 4^ o 3: o o o cc ^ 3 3 3 Sf 5 H 3 3- a 3 hi <-n ^ 3 Q O N ■ 5-3 tu ft O O :;• sr. c o c W 3 >< ►o „ SO 4> ~. Q v£) 4* SO 0\<-^ no, so b ~ oo oo o> o ~ T 1 « sb i» b »^ wM4» O OO Ox O O oo O On 00 "31 )S \) _ NO S-n N) O I- h vO lw W -Vj M 1-1 00 J- m O n^-- »5 r " J P ^ vf 00 OC-n n — NO - + O ooso 4^ ^vl so N On co O Og H - n 3- rt ft U 3 W 3- rr> 3 2.? fi N> <-n On 2 = m\oui oo O so cro O N N N On • O oo oo -f so no 5=» Cn On 0 ^ N ~ HT 4* O O -fr- 4> On ? S?5 9 5 "■ 3 3.. 5-2 3-|- 2- g-^ q O O O 4> O-l Nn 4». i— SO OSn or N «*» , NO ^ » K N ^ cJ 3 Hi OQC h O . 3- H 3^ r» as 3* s. s I " ft B "> v zr> Wo 3 g - m 3- < 3 ft f» a . •5.a O =' 3 ^" H ►» O D- « 4 . Cn 3 IS s>- 3 s^> 3S ml 55 Hi ! [ m ] To affift the imagination in forming an idea of the vaft diftances of the pla- nets from the Sun, let u« fuppofe, that a body projected from the Sun, mould continue to fly with the fwiftnefs of a cannon ball, viz. 480 miles every hour ; this body would reach the orbit of Mercury in 8 Julian years, 276 days; of Venus, in 16 years, 136 days; of the Earth in 22 years, 226 days ; of Mars, in 34. years, 170 days; of Jupiter, in 117 years, 234 days; and the orbit of Saturn, in 215 years and 286 days. If the reader jfhould think this idea too extenfive (notwithstanding its being a juft one) he may contract it in the following manner, which takes in both the proportional bulks and diftances of the Sun and Planets. The Dome of St. Paul's is 145 feet in diameter. Suppofe a globe of this iize to reprefent the Sun : then, a globe of 9 T 7 5 inches will reprefent Mercury ; one of 1 7 T 9 5 inches, Venus ; one of 1 8 inches, the Earth ; one of 5 inches di- ameter, the Moon (whofe diftance from 2 the [ i36 ] tnfe Earth is 240000 miles) <5ne of 10 inches, Mars; one of 15 feet, Jupiter; and one of n{ feet, Saturn, with his ring four feet broad, and at the fame diftance from his body, all around. In this proportion, fuppofe the Sun to be at St. Paul's ; then Mercury might be at the Tower of London; Venus at St. James's Palace; the Earth at Mary bone; Mars at Kenfington; Ju- piter at Hampton- Court, and Saturn at Cliefden : all moving round the Cupola of St. Paul's as their common center* A PRO- [ 137 ] A PROBLEM. I. Suppofe there are fix hands on the Dial- plate of a clock, all going round the fame way ; and that the firfl or flow - ejl hand, A, goes round in 34 hours ^ the next jhwejl hand, B, in 22 hours] the next , C, in 20 hours ; the next, D, in 18; the next, E, in 1 6 ; and the lqft y orfwiftefl, F, in 14. Hours : and that they all Jet off together, from a conjun&ion, at any given point of the Dial- plate. Qu. In how many hours afterward will they all he in conjunction again, and how many re- volutions will each hand have made in that time ? \ Let ^ c > d > be the periodical times or revolutions of A, B, C, D, E F, then, ^will be 24 hours, h 22 ] c 20, d 18, e it, and/ 14. I. The canon for finding the time that muft dapfe between the con- T junctions [ *38 ] jun&ions of A and £, is and con- fequently all its multiples; viz. If} tf} % Sec. on to m — u \ where m is any indefinite number of conjunctions. For, if A and B are in conjunction at the end of the time ~, 'tis evident they will be in conjunction again when as much more time has elapfed ; and fo on to infinity. 2. The canon for finding the times between the conjunctions of B and C is p. a nd all its multiples indefinitely ; And therefore, when or its mul- tiple, is equal to ^ or its multiple, A^ By and C will then be in conjunction again. For, by the firft expreflion, A and B will be in conjunction ; and by the fecond, B and C will be fo too. But the expreffions being equal, the times muft alfo be equal : that is, A, By and C will be in conjunction again. 3. The canon for finding the times between the conjunctions of C and D is [ *39 ] * s and all its multiples indefinitely, as above. And therefore, when any multiple of the conjunctions of A, By and C, is equal to any multiple of the conjunctions of Cand Z>, then Ay By C y and £), will be in conjunction again. 4. The canon for finding the times between the conjunctions of D and E is z~> an d a ^ its multiples ; and there- fore when any multiple of the con- junctions of Ay By C y and Z>, is equal to any multiple of the conjunctions of D » and E, then, A y B y C, D, and £, will be in conjunction again. 5. The canon for finding the times between the conjunctions of E, and F y 1S TZp an( 3 all its multiples indefinitely 5 and therefore, when any multiple of the conjunctions of Ay B y C, D, and £, is equal to any multiple of the conjunctions of £, and F, all the hands A, B, C, D y E y and F y will be again in conjunction. The multiples muft all be whole numbers, and the leaft that will do muft T 2 be [ *4° ] be taken, to find the times between the next fucceeding conjunctions. The times between the conjunctions of thefe fix hands, taking them by two and two, are as follows ; j^- b — 264 ; the number of hours in which A and B will come to their next conjunction, after their firft fetting out together. ~ — 220 : the number of hours in b — c 1 which B and Cwill come to their next conjunction, alter their firft fetting out together. — — 180 ; the number of hours in c — a J which C and D will come to their next conjunction, after their firft fetting out together. jzr =■ 144; the number of hours in which D and E will come to their next conjunction, after their firft fetting out together. And, i£ = 1 1 2 : the number of hours in f-j ■ - which E and F will come to their next conjunction j [ <4i ] conjunction, after their firft fetting out together. In working for fuch multiples (of integer numbers) as will make the above expreffions equal, in the leaft ratios of the times, I find they are as follows : 264 (or^) multiplied by 5, is equal to 220 (or multiplied by 6; equal to 1320 hours, for the time in which Ay By and C, will come to their next conjunction, after their firft fetting out together. And 1320 hours (the conjunction of A, By and C) multiplied by 3, is equal to 180 hours (or — ) multiplied by 22; equal to 3960 hours, for the time of the next conjunction of A y B y C, and Dy after their firft fetting out together. And 3960 hours (the Conjunction of A y By Cy and Dy) multiplied by 2, is equal to 144 hours multiplied by 55 ; (or ~) equal to 7920 hours, for the time of the next conjunction of Ay By C> [ J 42 ] Dj and E, after their firfl: fetting out together. And 7920 hours (thelaft mentioned con- junction) multiplied by 7, is equal to 112 hours (the conjunction of E y and F> or j£) multiplied by 495 ; equal to 55440 hours, the time in which all the fix hands, A, B, C, £>, E y and F y will be in conjunction again, after the inftant of their firfl: fetting out to- gether, from a conjunction at any given point of the dial-plate, and all moving round the fame way, in the times above mentioned. Now, as it will require 55440 hours (or 2310 days) to bring all thefe hands together again, after their firfl: fetting out together; divide 55440 hours by the number of hours in which each hand goes round, and the quotients will fhew that A has made 2310 re- volutions, B 2520, C 2772, D 3080, E 3465, and F3960. And, at the end of fo many more revolutions of each hand, they will all be in conjunction again ; and fo on continually. II. [ 143 ] II. the periodical times of the fix primary planets being given, andfuppofing them to have been all at once in a line of conjunction with the Sun ; to find how much time would elapfe before they were all in a line of conjuntlion with the Sun again. This Problem I had from Mr. Waring (now Profeffor of Ma- thematics in the Univerfity of Cam- bridge) in the year 1755. Let a, b, c> d> e, f be refpectively equal to the periodical times or revo- lutions of the fix planets about the Sun ; a being the longeft, or Saturn's period ; b the next longeft, or Jupiter's ; c the next, or Mars's ; d the next, or the Earth's ; e the next, or Venus's ; and / the fhorteft period of all, which is Mercury's : and let p, qp> rqp^ srqp, 8cc. be equal to the difference or time between the fucceeding conjunctions of any two, three, four, 6cc. of them. 6 'Tis t r 44 ] 'Tis evident that q y r 9 j, (the multipliers) muft be whole numbers, becaufe the numbers of conjunctions are fo. The time between the conjunctions of the firft two is ~ y that of the firft three is = q p (where n is any number aflumed, to make q a whole number) or, which is the fame, = ?> T^hrp bein g reduced to its loweft denominator, q will be equal to the numerator of that fradion. In the fame manner, r will be equal to the numerator of the fradion —"/ reduced to its loweft denominator • s will be equal to the numerator of the frac- tion 7~~ p reduced to its loweft de- nominator; and fo on, from the 'flowed to the quickeft revoking bodies in the fyftem : by which means, the times of all their conjunctions may be found. This C MS ] Tins Problem may be folved hy a differen t method^ as follows ; for which I a?n obliged to my getterous friend Mr* John Ford, Surgeon in BrifloL Let A, jB, C 3 D, £, i 7 , ftand for the fix planets, beginning with Saturn and ending with Mercury ; and a, b> c> d y e,f be the times of their periodical revolutions refpeclively. Then, by a known rule, the fynodical period, or conjunction, of A and 5, will be the time £^ ; and that of B and C will be P- ; That of C and D will be — v ; that v — c c — a ' of D and E will be i£ ; and that of E and F will be 4£ Now it is obvious, that A and 5 can never be in conjunction but in the time ~ , or fome multiple of it ; neither can B and C be in conjunction but in the time jp f , or fome multiple of that time. Z?, and C, will therefore be in con- junction when ~ is equal to where ^ and » reprefent two integer numbers, prime to each other ; which being re- U ipe&ively [ i 4 6 ] fpe&ively multiplied into ~ and ~ fhall make the two products equal. And thefe two numbers are eafily dif- covered ; for, as by fuppofition, ~ is equal to £-4, therefore, m.n : : : — S PC fr w — c a _ Reduce therefore an d — , into in- b—c a—b tegersof theleaftdimenfions, which fhall have the fame proportion to each other as thefe numbers have ; and you will have the multipliers m and n> and con- fequently the fynodical period or con- junction of Ay By and C\ which we fhall call R. In the fame manner may the fynodical period of C, D\ and E, be inveftigated, which call S: then find two prime numbers r and s in their loweft dimenlions, which fhall have the fame proportion to each other as the times R and S ; then will rS, or its equal oV, give the fynodical period, or conjunction of the five planets, Ay B> Cy Dy and £, which characterize by T. Find lafily, the fynodical period of E and jF, by the rule which de- note by X\ and the lean: integer num- bers [ r 47 ] bers t, X) in the fame proportion to each other as 7"* and X being found, tX or xT will be the fynodical period, or conjunction, of the fix primary pla- nets, A) B, 67, D, E y F ; or the time that muft elapfe between any conjunc- tion of them all, and the next fucceed- ing conjunction. Which time, being divided by the time of the periodical revolution of each planet, will fhew how many revolutions each planet has then made. There are feveral ways of finding the above-mentioned prime integer numbers or multipliers \ but the following is very convenient and eafy. Let i and - d be two of the fractions. Multiply the denominator of the firft into the numerator of the fecond, and vice verfd ; then ftrike out both the denominators, by which procefs the above fractions become ad and be ; which numbers are in the fame pro- portion as the fractions \ and, if they are prime to each other, are the num- bers required. But if they are not U 2 prime. [ '48 } prime, divide them by treiir greater! common divifor, in order tto reduce them to their lowed denomination. The reafon why thefe numbers muft be prime integers is plain : for,, if they were not fo, we mould not have the fynodic period required, buc fo me mul- tiple of it : and if they xvtrk not inte- gers, we fhould not have exact multi- JD 7 pies of the lower fynodic periods from which we deduce the higher. To facilitate calculations which may be made on thefe principles, I mall fubjoin the following Table, which fhews. the annual periods of the primary pla- nets, reduced to hours ; and their fy- nodical periods, taken two by two pro- greffively. But although the fynodical periods of the planets, taken two by two, is fo fhort, it muft not be ima- gined that the fynodical periods of three planets muft be proportionably fo too. The fynodic period of the Earth and Venus (by the Table) is i year, 218 days, 17 hours; and that of Venus and Mercury is 144. days, 12 hours;, but [ 149 ] but the fynodical period of thefe three planets is upwards of 5500 years. If the periods of three planets be fo incommenfurate, how much more fo muft be the periods of the fix revolving primaries of our fyftem? Indeed we here cannot but fee and admire the wifdom and providence of the Supreme Being I For, had the times of the an- nual revoliutions of the feveral planets been more commenfurate, the prefent arrangement of our fyftem would doubt- lefs have been greatly difturbed by the confpiring attraction of the fix bodies*, when they happened to be in conjunc- tion ; an arrangement, which, from the goodnefs of the Almighty, we muft conclude to be, in its prefent ftate, the beft adapted to anfwer the purpofes for which the fyftem was created. Names of the Their periodical revolutions Planets. reduced to hours. Saturn » ■ 258223^ zz a Jupiter ■ 103980 — b Mars 16487 — c Earth . 8766 = d Venus 53.93 & Mercury — . 2 1 1 1 =: / [ *5° 3 Their fynodical periods, or conjunctions with each other. Saturn and 7 Y - D- H. Hours. Jupiter 1 19 313 10=174076- — a — b Tupiter and \ ■' 0 ' be Mars $ 2 ^ 21 = l ^9^ c Mars and } 0 _ 49 I0 — 18710=-!^ the Earth 3 c—d Earth and ? 0 de Venus I 1 218 17= 14015=^ Venus and ? o 144 12= 346^=-^ Mercury $ e ~f To illuftrate the ufe of this Table, let it be required to find the fynodical period or conjunction of the Earth, Venus, and Mercury. That of the Earth and Venus 140 15 hours, = and that of Venus and Mercury is 3468 hours, m ~- f , Therefore, from what has been al- ready laid down (fee page 145) the fy- nodical period of the three planets will be when m x 140 15 is equal to nx 3468; or when #2: n:: 3468 : 14015; m and n being the leaft integer numbers in the proportion of 3468 to 140 15. But thefe numbers being integers, and C r 5i ] in their loweft terms already, they require no reduction. Therefore, 3468 x 1 401 5 give the fynodical pe- riod of the three planets, ss 48604020 hours; =z 5544. years, 221 days, 12 hours. The reader may proceed to find put the fynodic periods or conjunctions of the reft, according to the foregoing rules. The following problem is x>f the fame nature with this ; but, as it is more familiar and obvious, it may better ferve to confirm the truth of the method we have ufed to inveftigate the fynodical period of bodies revolving the fame way, but in different times, about the fame common center. III. guppofe the hour > minute ', and fecond hands of a Clock to be in conjun&ion at the hour of XII. It is required to find when they will be in ^junEi ion again? Here we have the periodical revo- lution of the hour-hand =720 minutes, ~- a\ the periodical revolution of the 1 minute- [ *S* ] minute-hand = 60 minutes, = h\ and that of the fecond-hand = 1 minute, = c: from whence we collect ~j =^min. =^°=^>in. for the fyno- bbo 1 10 II J dical period or conjunction of the hour and minute hands ; and £~ == — for the fynodical period of the minute and fe- cond hands. Then, to find the fyno- dical period of all the three hands, we muft (as in the above Problem) fuppofe f !t2S2I2 — f L^2* from whence we have 11 59 » m 1 n : : w : 7T # Now, the leaft integer numbers, reprefented by m and tz, in the proportion of ^ to ~ are 1 1 and 708. Therefore 11 : 708 — and confequently ii^2 ( = 708 x|) = the fynodical period of the three hands of the Clock ; =720 minutes, or juft 12 hours* The periodical revolutions of the Sun and Moon round the Ecliptic, and their fynodical periods or conjunctions with each other, may be familiarly re- prefented by the motions of the hour and C *53 ] and minute hands of a watch, round its Dial-plate. For, the Dial-plate is divided into 1 2 hours, as the Ecliptic- is divided into 1 2 figns ; the hour-hand goes round in 1 2 hours, as the Sun does in 12 months, and the minute hand goes round in 1 hour, as the Moon does in (fomewhat lefs than) a month. And, as the Moon is never in conjunc- tion with the Sun in that point of the Ecliptic where (he was at the laft con- junction before, fo the minute-hand never is in conjunction with the hour- hand at that point of the Dial- plate where it was at the laft preceding con- junction. So that, the 1 2 hours on the Dial-plate may reprefent the 1 2 figns of the Ecliptic ; the hour-hand the Sun, and the minute-hand the Moon : only, the motion of the minute-hand is too flow for the Moon in proportion to that of the hour-hand compared with the motion of the Sun. For, in the time of the Sun's going round the Ecliptic, which is 1 2 calendar months, there- are 12,36 conjunctions of the X Sun [ 154 ] Sun and Moon ; but in the time the hour-hand goes round the Dial-plate, the minute-hand is only 1 1 times con- joined with it. Thefe hands are always in conjunction at XII o'clock. The firft column of the Table fhews the number of their conjunctions in i 2 hours, and the col- lateral lines fhew in how many hours, minutes, &c. after XII, they come to their fucceeding conjunctions marked in the firft column ; the time between any conjunction and the next being i hour, 5 minutes. n o ,3 I 2 3 4 5 6 7 8 9 io i i Hou. m. 11 in IV V VI VII VIII IX X I 5 27 16 21 49 5 27 16 21 49^ II IO 54 32 43 38 10 54 32 43 3 8 -r^ III 16 21 49 5 27 16 21 49 5 27-A- mi 21 49 5 27 16 21 49 5 27 16^ V 2 7 16 21 49 5 27 16 21 49 5rV VI 32 43 3 8 IG 54 32 43 38 10 54tt VII 10 54 32 43 38 10 54 32 43tV VIII 43 38 10 54 3 a 43 38 10 54 32-rV IX 49 5 27 16 21 49 5 27 16 21A X 54 32 43 38 ,10 54 32 43 38 XII 0 0 0 0 0 0 0 0 0 O If the above procefs was carried on to infinity, in the horizontal lines, the numbers would circulate at every fixth column. To [ «S5 ] To reprefent the motions of Jupiter s four Satellites round Jupiter ', in a clock ; and Jhew the times of their Eclipjes in Jupiter s Jhadow* On four hollow arbors^ let there be four bent wires of different lengths, to carry the Satellites round Jupiter, as the arbors are turned round within one another; and let Jupiter be fixed on the top of a folid axis or fpindle, on which all the arbors are turned round ; the wires being fo bent, as that the Satellites, on their tops, may be of the fame height with Jupiter's ball. The diameters of the Satellites fhould not be above a lixth or feventh part of the diameter of Jupiter; and, to be at their proper diftances from him, the diftance of the neareft Satellite fhould be 5| femidiameters of Jupiter from his center; the fecond " Satellite, 9 fe- midiameters of Jupiter diftant from his center; the third femidiameters; and the fourth, 25I of his femidiame- ters from his center. X 2 3Let [ *S6 ] Let four wheels of different fizes, and different numbers of teeth, be fixed upon the lower end of the abovemen- tioned arbors, in a conical manner, as defcribed in the former machine (pag. i 29); the wheel on the fmalleft arbor, that carries the firft Satellite, having 22 teeth; the wheel on the next arbor, that carries the fecond Satellite, 33 teeth; the next bigger wheel on the arbor that carries the third Satellite, 43 teeth; and the largeft wheel of all, on the arbor that carries the fourth (or outermoft Satellite), 67 : the biggeft wheel being the uppermoft, and the fmalleft the lowermofL Thefe four wheels muft be turned by other four, all fixt on a folid axis, in an inverted conical manner, with refpecl; to the former wheels on the hollow arbors; and then, all the four on the folid axis will be turned round in one and the fame time. The fmalleft wheel (or uppermoft one) on this axis 5 muft have 38 teeth; and [ *57 ] and turn the wheel of 67 teeth, which carries the fourth Satellite. The next wheel on the axis muft have 42 teeth ; and turn the wheel of 43 teeth, which carries the third Sa- tellite. The next bigger wheel below, on the axis, muft have 65 teeth; and turn the wheel of 3 3 teeth, which car- ries the fecond Satellite. And, The lowermoft, and biggeft wheel on the axis, muft have 87 teeth; and turn the wheel of 22, which carries the fir ft Satellite. Then, If the clock turns the folid axis with all its wheels round in 7 days, the fir ft Satellite will be carried round Jupiter in 1 day, 18 hours, 28 minutes, 57 feconds ; the fecond Satellite, in 3 days, X 3 hours, 1 7 minutes, 46 feconds ; the third, in 7 days, 3 hours, 59 minutes, 54 feconds ; and the fourth Satellite in 1 6 days, 1 8 hours, o minutes, o feconds ; which agrees fo nearly with their re- volutions in the heavens, as not to differ fcofibly, in a long time, from them. ? An£ C '58 1 And then, if a piece of black wood be turned, a little conical in its fhape, having its thickeft end as broad as the diameter of Jupiter is long, and be made hollow to fix on che back of Ju- piter, and have notches cut in it for the Satellites to pafs tkough : it will reprefent Jupiter's fhadow; and when the Satellites are in the notches, it will fliew them to be eclipfed. The times of the immerfions of the Satellites of Jupiter in;o his fhadow, or of their emerfions from it, may be had from White z Epheraeris every year; and if the Satellites are once put juft entering into the notches for the im- merfions, or juft leaving it for the emer- fions, at the proper times by the clock ; they will keep right to the times thereof for more than a year afterward, without needing any new adjuftment. And, in order that they may be fo fet, without afTeding the wheels that move them, their wires fhould be fixed into round collars, which go moderately tight on the tops of the four hollow arbors, fo as [ 159] as they may be carried about Jupiter by the tightnefs of the collars ; and yet at any time may be moved, and fet right by hand. All the numbers of teeth in the wheels' are here copied from Mr. Ro- mer\ Satellite inftrument, except thole for the fecond Satellite; where Mr. Romer has a wheel of 63 teeth, turning a wheel of 3 2 : inftead of which, I make a wheel of 65 turn a wheel of 33, which comes much nearer the truth. About 16 years ago, I made one of thefe inftruments, to be turned with a winch by hand. It had a Dial-plate divided into the months and days of the year, within which was a circle di- vided into twice twelve hours. On this Plate there were two indexes, one of which was moved round, over all the 365 days of the annual circle, in 365 turns of the winch : and the other index was moved round, over all the 24 hours, in one turn of the winch ; by which means I could, in a very Ihort time, fhew at what times of the days [i6o] " days the Satellites would be eclipfec^ throughout the whole year. And, after having the above numbers for the mo- tions of the Satellites, any Clock-maker may eafily conftrud a machine of this fort ; by which, the times of the Im~ merfions or Emerfions of the Satellites may be known before-hand, in order to be prepared for obferving them m the heavens. How [ i6i ] How to confiruEi an Orrery for jhewing the annual revolutions of Mercury ', VenuS) and the Earthy round the Sun y in their proper periodical times ; the Moons motion round the Earthy and round her own axis, with all her dif- ferent Phafes : the motions of the Sun 9 FenuSj and the Earth, round their r effective axes ; the vicijfiudes of Sea- fons y the retrograde motion of the nodes of the Moons Orbit > with the times of all the New and Full Moons, and of all the Solar and Lunar Eclipfes* Let a wheel of 6.12 inches* having 74 teeth, be fixed on the axis of the handle or winch, and turn a wheel of 1.80 inches diameter, having 32 teeth, which turns a wheel of 7.3 teeth, whofe diameter is 6.11 inches; and, on the axis of this laft wheel let there be one of 32 teeth, of 1.80 inches diameter, turning a wheel of 160 teeth, whofe diameter is 8*97 inches; and that wheel to turn two wheels of 32 teeth Y each, C '62 ] each, and diameter 1.80 inches. One of thefe laft wheels of 32 to have a fmall wheel of 16 teeth on the top of its axis, turning another of the fame number and fize, and that one to turn fuch another, on the top of whofe axis (inclining 23^ degrees) is the Earth, which turns round in the fame time as the winch ; each turn anfwering to 24 hours. The Earth is covered half over with a black cap, and turns freely round within the cap, whofe edge re- prefents the boundary of light and darknefs, and fhews the times of the [apparent] rifing and fetting of the Sun, as the different places of the Earth emerge from below it, or go in under it. The other wheel of 3 2 teeth (turned by the forefaid wheel of 160) has an index on its axis, which goes round a Dial-plate of 24 hours, in the time the Earth turns round its axis. The fame wheel of 32 turns one of 64 teeth, whofe diameter is 3.60 inches, and turns a wheel of 30 teeth, 1.6 inches diameter ; [ 1 63 ] diameter ; and on the axis of this wheel is a fingle threaded fcrew, turning a wheel of 63 teeth, whofe diameter is 3 inches, and turns a wheel of 24 teeth, 1.23 inches diameter, which turns a wheel of 63 teeth, 3 inches diameter; which laft wheel carries the Moon round her orbit iri 27 days, 7 hours, 43 mi- nutes, and from change to change, in 29 days, 12 hours, 45 minutes- The firft wheel of 63 teeth has an index on the top of its axis, which goes round a circle divided into 2 9^ equal parts in the time of a Lunation ; and mews the Moon's age every day. A fmall wheel of 20 teeth is fixed on a focket, among the other work, below the Earth ; and by the bar that carries the Moon, a wheel hanging on the bar, of 20 teeth, turns another of the fame number and fize, on whofe hollow axis is the Moon's black cap, which always faces the Sun, and fhews the Moon's phafes, as me turns round her axis, which is within the hollow axis of her cap, Y 2 On [ ?6 4 ] On the axis of the wineh is a pinion of 8 leaves, turning a wheel of 25 teeth, which turns another of the fame num- ber, on whofe axis is a pinion of 7 leaves, ~ parts of an inch in diameter, which turns a wheel of 69 teeth, whofe diameter is 4.12 inches, and has a pi- nion of 7 leaves on its axis, turning a wheel of 83 teeth, which is fixed to a frame that contains feveral of the above- mentioned wheels within it, and carries the Earth round the Sun in 365 days, 5 hours, 48 minutes, 57 feconds. The .diameter of the wheel of 83 teeth is 6.12 inches. On the axis of the laft mentioned wheel of 69 teeth, is a pinion of 10 leaves, turning a wheel of 73 teeth, whofe diameter is 5.82 inches, and is fixed to a frame in which are feveral pther wheels (to be defcribed by and by) and carries Venus round the Sun in 225 days, 17 hours. On the axis of the forefaid wheel of §9 teeth is a wheel of 78, whofe dia- meter is 3.68 inches, and turns a wheel of [ '65 ] of 64 teeth, whofe diameter is 2.3 in* ches, and on the top of whofe axis the Sun is placed ; the axis inclining j\ de- grees, and the Sun turning round by it in 25 days, 6 hours. Iti the center of the machine, below the Sun, there are three wheels fixed on the ftem, round which the whole work moves ; the ftem itfelf being fixed into the bottom of the box which contains the work. The lowermoft of thefe three wheels is 2.95 inches in diameter, and contains 50 teeth, which take into the teeth of another wheel of the fame number and fize, and this laft wheel takes into the teeth of another of the fame number and fize, for keeping the parallelifm of the Earth's axis in its whole courfe round the Sun ; on which parallelifm, the whole variety of the Seafbns depend. On the axis of the middlemoft of thefe three wheels of 50 teeth is a wheel of 59, (a little bigger than the wheel of 50) which takes into a wheel of 56 teeth (of a fomewhat fmaller fize) 3 and [ 166 ] and this wheel of 56 moves the Nodes of the Moon's orbit backward, through all the figns and degrees of the Ecliptic in 1 8^ years. Above the fixed wheel on the middle ftern, of 50 teeth, is a fixed wheel of 74, whofe diameter is 6,12 inches, and takes into a pinion of 8 leaves, on the top of whofe axis is a fmall wheel of 16 teeth, turning another of the fame num- ber and fize, and that turning another of the fame number and fizealfo, on whofe axis, inclining 7 5 degrees, Venus is turn- ed round in 24 days, 8 hours; which is her diurnal period, according to Bian- chints obfer vat ions. Above the faid wheel of 74 teeth, and fixed on the fame ftem, is a wheel of 28, whofe diameter is 1.74 inches; this wheel takes into the teeth of ano- ther of the fame number and fize, which takes into a third of the fame number and fize alfo; and this third wheel keeps the parallelifm of Venus's axis throughout her whole annual pe- riod round the Sun. On [ l6 7 ] On the axis of the middlemen: of thefe three wheels of 28 teeth is ano- ther of the fame number and fize, which turns a wheel of 1 8 teeth, whofe diameter is 1 . 1 2 inches, and which turns another of the fame number and fize, which carries Mercury round the Sun in 87 days, 23 hours. Any perfon who is not accuftomed to the making of Orreries may perhaps be apt to think, that all the abovemen- tioned motions might be performed by fewer wheels ; and an expert Clock- maker, by computing the periodi- cal times of the planets revolutions from the numbers of teeth in thefe wheels, might pronounce them to be very inaccurate. But it ought to be confidered, that there is a very great difference between the rotations of wheels which always keep in the fame places, and of thofe which do not only turn round, but are alfo carried round others, continually changing their places and pofitions. As I wanted an Orrery more exacl in the annual periods of the pla- nets, c m ] nets, and motion of the Moon round her orbit, than any one I have yet feen ; the common Orreries being more adapt- ed for reading public lectures upon; where it is fufficient to mew and explain the general phenomena; the makers generally content themfelves with ha- ving fuch numbers as will carry the Earth round the Sun in 365 of its di- urnal rotations, the Moon round from change to change in 29^ days, and her nodes round the Ecliptic in 19 years; I have taken the pains to calculate the abovementioned numbers, which are far more exact ; and got a good workman to make an Orrery under my infpection, in which the diameters of the wheels, and their numbers of teeth are exactly defcribed ; and which I now give freely to thofe who choofe to work by them. Another C * 6 9 ] Another Orrery. About twelve years ago, I made a large wooden Orrery, for (hewing only the motions of the Earth and Moon, with the retrograde motions of her nodes, and the phenomena arifing from all their motions. The Earth had not its diurnal and annual motions carried on by means of a winch, but by hand j and as the Earth was moved round the Sun, the Moon was carried round the Earth in her orbit, and her nodes had their retrograde motions. As there is fomething very particular and fimple in the conftru&ion of this machine ; and as the Moon's motion in it will not vary above one degree from the truth in 304 years; and as it anfwers as well in Leap years as in common years ; and has only feven wheels and one pinion in it ; I fhall here mention its ufe, but mull beg to be excufed from defcribing the po- fition of its wheels, and their numbers of teeth, becaufe I intend to inftrutf: my Z fon, [ *7° ] fon if both he and I live till the pro- per time, how to make it for his own benefit. Befides, it would be very dif- ficult to make it intelligible by a de- fcription, without feeing it ; efpecially as fome of the wheels are not only di- vided into very uncommon numbers of teeth, but alfo that, in fome of the wheels, equal numbers of teeth are con- tained in unequal fpaces ; for mewing the inequality of the Earth's annual mo- tion round the Sun, and of the Moon's motion round her orbit. It fhews the following matters very readily. The lengths of days and nights at all places of the Earth, and at all times of the year ; with all the viciffitudes of Seafons. The Sun's place in the Eclip- tic on any given day of the year, and time of the day ; with his Declination, Altitude, and Azimuth at any time ; alfo his Amplitude, and the time of his rifing and letting. The time of the day, by the obferved Altitude or Azi- muth of the Sun. The variation of the compafs, in any place, whofe latitude is 3 known [ m 1 known by a fingle observation of the Sun's Altitude, taken at any time, either in the Forenoon or Afternoon. The Moon's periodical and fynodical revo- lution, with her rotation on her axis, and different phafes. The retrograde motion of her Nodes, and direct motion of her Apogee. Her mean Anomaly and elliptic Equation, by which her true place in her Orbit is very nearly found at any time. Her Latitude, De- clination, Altitude, and Azimuth, at any time when fhe is above the hori- zon. Her Amplitude, and the time of her rifing and fetting, however affected by her Latitude. The times of all the New and Full Moons, and of all the Solar and Lunar Eclipfes, within the limits of 6000 years before or after the Chriftian iEra; with an eafy method of rectfiying the Machine, in lefs 1 than two minutes of time, for the beginning of any given year within thefe limits : and when it is once rectified, it will keep right for 304 years either back- wards or forwards ; at the end of which Z 2 time, [ *72 ] time, the Moon muft be fet one degree forward in her orbit. The fmall dif- ference in the time of the Moon's ri- ling, in Harveft, throughout the week in which fhe is Full ; and the great dif- ference in the time of her fetting during that week. The Receffion of the Equi^ no&ial points, in the Ecliptic. The Phenomena of the Tides, and the caufes of many apparent irregularities in their heights, and times of ebbing and flowing. *fhe Mechanical Paradox* This is a fmall kind of Orrery, which I contrived and made about fifteen years ago. It has only five wheels, and fhews the Seafons, the retrograde motion of the Moon's Nodes, and the mean times of Eclipfes of the Sun and Moon. I gave it the above name, becaufe there is one wheel in it as thick as three of the others ; and that wheel takes fairly and equally deep into the teeth of thefe three other wheels (which are quite indepen- dent [ 173 ] dent of, and unconnected with, each other) ; and yet, the thick wheel affects the three wheels in fuch a manner, and at the fame time, as to turn the uppermoft of them forward, the middlemoft back- ward, and the lowermoft no way at all. For a Copper-plate of this machine, with a printed defcription in which the paradox is folved, I refer the reader to a Shilling Pamphlet, fold at Mr. Cadelfs Shop, oppofite Catherine Street in the Strand, Afiort account of the Silk Mills at Derby. In thefe Mills are 26586 wheels, and 97746 movements, continually working except on Sundays. This grand ma- chine is difpofed in four ftories of large rooms above one another; and the whole is actuated by one great Water- wheel, which goes round three times in a minute. In each time of its going round, 73728 yards of Silk are twifted : fo that, in 24 hours, 3 18504960 yards are [ *74 ] are twifted. The water-wheel is kept conftanrly going ; but on Sundays it is difengaged from all the reft of the work. Any part of thefe movements may be ftopt without the leaft prejudice or in- terruption to the reft. TVond'rcus Machine ! Thy curious Fabric (hews How far the power of human wifdom goes ! Where many thoufand movements all attend Upon a wheel, and on that Caufe depend. Sceptic, advance ! propofe thy fcheme of wit. That faith to reafon always rauft fubmit. Whence learn'd thefe movements to obey command ? Who taught them how to roll, and when to ftand ? Was it by chance this curious fabric came ? Or did fome thought precede, and rule the Frame? Worthy the Mortal, on whofe Soul, confeft, His Great Creator's Image ftands impreft! Now turn from Earth to Heaven thy doubting eyes, And read th! amazing Glories of the Skies ! Worlds without number roll in different Spheres, Keep to their Seafons and complete their years. Five thcjufand circuits, made with equal force, The Earth has finifh'd by its annual Courfe. The Sun difpenfes beams of genial Light, And lends his rays to cheer the gloomy night. Stupendous Power and Thought! Enquire no more : Own the First Mover $ and, convinc'd, adore. Rules [ '75 ] Rules for finding the correfponding years of the Julian Period 'with the years of the worlds and years before and fince the birth of Christ ; fuppofeng (with Mr, Bedford, in his Scripture Chro- nology) that the Creation of the World was in the 7 06 th year of the Julia7t Period ; and that the birth of Curjst was {according to the vulgar /Era thereof) in the 4.713th year of the Julian Period. From any given year of the Julian period fubtradl 706, and the remainder will be the years of the world's age. If the number of the given year of the Julian period be lefs than 4713, fubtradt it from 4713; and the remain- der will be the number of years before the year of Chrift's birth. If the given year of the Julian pe- riod is greater than 3967, fubtrad 3967 from it ; and the remainder will be the number of years after the famous iEra of Nabonaffar. Subtrad [ i?6 ] Subtract i from any given year of the Julian period, and divide the re- mainder by 4; if nothing remains, the given year is a Leap year : but if 1, 2, or 3 remains, it is the firft, fecond, or third year after Leap year, in the Old Stile. If any year before the year of ChrhTs birth be given, fubtract its number from 4713, and the remainder will be the year of the Julian period. And if you fubtract the faid given year from 4007, the remainder will be the years of the world's age. If any year after the year of Chrift's birth be given, add 4713 to it, and the fum will be the year of the Julian pe- riod ; or if you add 4007 to it, the fum will be the years of the world's age. If any year of the world's age is given, add 706 to it, and the fum will be the year of the Julian period. If the given year of the world be left than 4007, fubtracl it from 4007; and the remainder will be the number of C *77 ] of years before the year of Chrift's birth. But, if the given year of the World be more than 4007, fubtrad 4007 from it ; and the remainder will be the number of years after the year of Chrift's birth. A Table of remarkable JEras and Events: 9- 10. 11. 12. 13. »4« 15- 16. 17- |8. The Creation of the World The Flood — — The AJJyrian monarchy founded by Nimrod • — — — The birth of Abraham — — The destruction of Sodom and Gomorrah — The kingdom of Athens founded bv Cecrops — ■ - Mofes receives the ten command- ments from God ■ — The Ifradites enter Canaan The deitruftion of Troy The beginning of king David's reign The founding of Solomon's The Argonautic Expedition Lycurgus formed h s excellent Arbaces, firft king of the Medes Mandaucus, the fecond — Sofarmus, the third ■ The beginning of the Qregi Olympiads ' « Artka, the fourth king of the A a Julian Period. 706 2362 World's Age. 0 1656 Before Chrift. 4007 2351 2537 2714 1 8- I 2008 2176 1999 2816 21 10 1897 3157 2451 1556 3222 3262 3529 2516 2556 2823 149! 1451 1184 3650 2944 1063 37°I 377 6 2995 307P 1012 937 3829 3838 3865 39. J 5 3103 3*3 2 3159 3029 884 875 848 798 3938 3232 775 3945 3259 768 [ i78 ] ig. The CatonlanEpocha of the build- ing of Rome — | — — 20. The J&ra of Nabonajdp — 21. The deftruclion ot Samaria by Salmanefer ■ ■ 22. The ftrft Ecllpfe of the Moon on record 23. Car dice a, the fifth king of the Medes 24.. Phraortes, the fixth 25. Cyaxares, the feventh — — 26. The. hilt Babylonijh captivity by Nebuchadnezzar 4 23 3 3' The long war' ended between the Medes and Lydians — — The fecond Babyloni/b captivity, and birth of Cyrus The dsftruchon of Solomon's Temple Nebuchadnezzar ftruck with mad- Daniel's Vifion of the four mo- narchies — — - 32. Cyrus begins to reign ■ — 33. The battle of Marathon — 34. /rtaxerxes Longimanus begins to reign The beginning of Daniel's fe- 35- 36. venty weeks of years The beginning of the Pelopon- rejian war • 37. Alexander's victory at Arbela 38. His death * 39. The captivity of roc, 000 Jews by king Ptolemy The Coloffus of Rhodes thrown down by an earthquake — dntiochus defeated by Ptolemy Philopater The famous Archimedes mur- dered at Syracufe 40. 41. 42. 43- 44. Jafon butchered the inhabitants 01 Jerufalem — — Corinth taken and plundered by Conful Mummius • — - Julian "erio; . World's; Age. Before Chrift. 39 61 39-7 3 2( ;5 3261 752 746 399 7 - 3286 721 3993 3287 720 399 6 4058 4080 3290 3352 ' 3374 7*7 655 6 33 4107 3401 6c6 41 1 1 34 c 5 602 4114 3408 599 4125 3419 588 4141 3438 569 4158 4177 4 22 3 345 2 347 1 3517 555 536 490 4 -'49 3543 464 4256 355° 457 4282 43 83 4390 3576 3677 3684 43 1 33° 3 2 3 4393 3687 320 4tQI TT.7 3785 . 222 4| 9 6 3700 217 45 06 3800 207 4543 3337 ! I7O 45 6 7 45 I46 . Juliu [ *79 3 4$. Julius Cafar invades Britain 46. He corrects the calendar 47. Is killed in the Sena te-houfe 48. Herod made king of Judea 49. The battle at Adiuni — ,50. Agrippa builds the Pantheon at Rome • - — - — 51. The true JEra of Christ's 52. The death of Herod S3- 54. 55- 56. 57- 58. 59- to. 61. 62. 63. 64. 65. 66 6 7 . Julian Feiiod, 4659 4607 4671 4 6 73 4683 4668 4709 4710 The Dionyjian, or vulgar JEra of Ch rist's birth • The true year of his Cruci- fixion > The deftrudtion of Jerufakm Adrian built the Jong wall in Britain — . . • Conftantius defeated the Pids in Britain • • The council of Nice • The death of Confiantine the Great - The Saxons invited into Britain The Arabian Hegira, or flight of Mohammed — — — The death of Moha?nmed — The Perjian Tefdegird The art of Printing difcovered The Reformation begun by Martin Luther Oliver Cromwell died — — Sir Isaac Newton born at Wcoljlrope in Lincobijhire, De- cember 25 ■ — ■ — went to Trinity College in Cam- bridge • ■ — — was elected Fellow of that Col- lege _ — invented the Fluxions — made ProfeiTor of Mathematics, in the room of Dr. Barroip — p'ublifhed his Princip ia — exerted himfelf for Religion A a 2 47^ 4746 4783 4833 5019 S°3 8 5050 5158 5 335 5343 5344 6153 6230 6371 6355 6373 6380 638; 6382 6400 6401 World's Age. 39 v3 3961 39 6 5 39 6 7 3977 Before Chrilt. 54 46 4.2 40 30 3982 2> 4003 4004 4 3 Since Chnft- 4007 0 4040 4077 33 70 4 I2 7 120 4313 433 2 306 725 4344 4452 337 445 4° 2 9 4 6 37 4638 5447 002 630 631 1440 55 2 4 5665 1517 1658 5649 1642 rfi6-7 1 660 5 6 74 5676 1667 1669 5676 5684 I 5 6 *5 1 669 1687 1688 C 'So ] Sir Isaac NeWton made Prefident of the Royal Society Knighted by Queen Anne — • died, March 20 ■ 1 In this Table, the years both before and fince Chr reckoned exclusive from the year of his birth. ' Julian Period, 6416 6418 6446 World's Since Age. Chnft. 5700 1703 5702 1705 5734 1727 He The year of our Saviour's Crucifixion ascertained \ and the darknefs at th% time of his Crucifixion proved to be fiipernaturaL Concerning the time of our Saviour's entering upon his public miniftry (which maybe called the time of his appearance, becaufe, till then, he was not publicly known, fo as to be talked of) and alio concerning the time of his death, there is a very remarkable prophecy in the IXth chapter of the book of Daniel^ from the 24th verfe to the end; which is in our Enojifh Tranflation as follows: Ver. 24. Seventy we eh are determin- ed upon thy people, and upon thy holy city, to finifh the tranfgreffion, and to make an end of fins, and to make recon- ciliation for iniquity, and to bring in ever- lafti?2g righteoufnefs, and to feal up the vifio7i and prophecy and to anoint the mofil holy, 2 5 . Know therefore and underfiand^ that fro?n the going forth of the command- i 182 ] commandment to rejlore and build Je~ rufalem^ unto the Mejfah the prince^ pall be feven weeks ; and three/core and two weeks the fireet Jkall be built again^ and the wall y even in troublous times. 26. And after threefcore and two. weeks pall Mejfiah be cut off y but not for himfelf: and the people of the prince that pall come, pall dejlroy the city and the fan8luary y and the e7id thereof pall be with a food, and unto the end of the war defolations are determined, 27. And he pall co?ifrm the covenant with many for one week : and in the midjl of the week he pall caufe the facrifices and oblations to ceafe y a?id for the over- fpreading of abomination he pall make it deflate^ even until the confummaticn> and that determined^ pall be poured upon the deflate. In the above tranflation, one part of the 25 th verfe is moft injudicioufly pointed with a femi-colon at feven weeks ; which ought to run thus., feven weeks and' threefcore and two weeks. In the 24th verfe, what we have ren- dered c ] dered prophecy ; , kprophetin the original: and in fome tranflations, which 1 have procured from thofe who underftand the Hebrew very well, inftead of vijion and prophecy \ it is rendered vifions and pro- phets. In ver. 27. where we have it the midjl of the week, all the tranflations I have procured render it the half part of the week ; which may be taken either for the firft or laft half part of it. In the fame verfe, where we have it And he floall confirm the covenant with many for one week ; fome tranflations render it And in one week a covenant fhall he confirmed with many. Now let the whole be put together agreeable to this tranflation, without dividing it into different verfes (which is only of modern invention) but pointing it here and there for the fake of reading ; and it will run thus : Seventy weeh* are determined upon thy people * Seventy /evens* according to Mr. Purver's tran- flation j which may be reckoned fevens of years sa well t llf ] people 7 ' T 7 T r f ^ 00 ^ of fms\ and reconciliation for iniquity, and to iring in ever lofting right eoufnefs, and ■to feal up the vifions and prophets, and to anoint the mojl holy f . Knew therefore, and uftderfland, that from the going forth of the commandment to reft ore and build Jerufalem, unto the Meffiah the prince, fhall be feven weeks and threefcore and two weeks : the Jlreet fthall he built again, and the wall even in % troublous times. And after threefcore and two weeks fhall Meffiah be cut off, but not for himfelf. ( A?td the people of the prince that fall come fhall defroy the city and f ancillary, and the end thereof fhall be with a floods and unto the end of the war deflations are determined.) And in one week a covenant fhall be confirmed well as fevens of days : and in the 6th verfe of the 4th chapter of Ezekiel, we have thefe remarkable words of God to that Prophet : " / have appointed thee each day for a year." -f* Some tranflate this, the holy of holies^ and Mr, Purveiy the very holy one. J By moft tranflators, in the fir aitnefs of times, with with many j and in half part of the week he || Jhall abolijh the facrifces and offer- ings. And for the overfpreading * of abominations he Jhall make deflate even unto confuming ; and that which is deter- mined Jhall be poured upon the deflate. 'Tis evident, that the firft part of this prophecy relates to the coming of Chrift ; to his being put to death, not for himfelf but for the fins of mankind, by which great facrifice he was to put an end to all other facrifices and of- ferings ; to his introducing the righte- oufnefs of ages, and fealingup (or putting an end to) prophecies. And that the latter part mentions the deftruction of Jerufalem, in a very finking manner. In the feventh chapter of Ezra, we have an account of a very ample com- million (or commandment) which was given by king Artaxerxes (who was called Artaxerxes Longimanus) to Ezra, to go up to Jerufalem, in order to repair that city, and reftore the ftate of the H The Meffiah. * Wing in the Hebrew. B b Jews ; [ i86 ] Jews ; and that Ezra took his journey on the firft day of the firfl: month, viz. the month Nifan ; which began about the time of the vernal equinox. And on the 14th day of that month (reckoned from the New Moon, at which the month began) the Paflbver was always kept; for Jofephus* expreffly fays, " The paffover was kept on the . But, [ i8 7 ] But, if we count many revolutions of 70 common weeks, from the time of the Jewifh paffover in the 457th year before the vulgar aera of Chrift's birth, we mall find that no Meffiah or Savi- our did appear on the Earth within that fpace of time : nor will thefe reckonings lead us from one Paffover to another. And it is certain, from the four Gofpels, that Chrift was crucified at the time of the Paflbver ; and St. John, chap, xviii. ver. 28. is fo particular, as to inform us that our Saviour was crucified on the very day that the Paffover was to be eaten by the Jews, who would not de- file themfelves by mixing with the mul- titude early in the morning, at the time of his trial. From thefe circumftan- ces it is plain that thefe prophetic weeks mean fomething very different from the weeks by which we commonly reckon. In the Old Teftament, we read of weeks of years, as well as weeks of days. For, as every feventh day was to be a fabbath for man, on which he was to reft from his labour ; fo every feventh B b 2 year [ "88 ] year was to be a fabbath for the land, in which it was to reft from tillage. Let us therefore take thefe 70 weeks to be weeks ox years, making 490 years in all ; and the reckoning will lead us from the Paffover in the 4157th year before the year of Chrift's birth to the Paflbver in the 3 3d year of our Saviour's age, accounted from the vulgar aera of his birth. It is expreffly foretold in this pro- phecy, that from the time of the com- mandment's being given to reftore and build Jerufalem, to the Meffiah the prince, (or to the time of his appear- ing in his public character) there mould be feven weeks and threefcore and two weeks; or 69 weeks in all: the firft feven of which, being the ftraiteft or fhorteft of the times, confifting of 49 years, we may very well allot to the repairing of Jerufalem ; after which, there mould be threefcore and two weeks, or 434 years, to the public ap- pearance of the Meffiah: and then there remained only one week, or feven years. [ *s 9 3 years, for the public miniftiy ; which, I apprehend, is meant by confirming the covenant with many. But as fome of the Tranflations which I have procured, fay, concerning that week, And in one week a covenant Jhall be coitfirmed with many ; and all of them have it % and in half part of the week (which might be either the firft or laft half of it) he jhall abolijh the facrifices and offerings , it does not appear that the Median is brought in for the whole of the feventieth week, but only for one half of it, in confirming (or eftablifhing) the new and everlafting covenant of the Gofpel ; by which, the righteoufnefs of agesy mentioned in the firft verfe of the prophecy , feems to be plainly meant. And when we confider, that Christ's meffenger, John the Baptijft, preached fo long before Chrift took the public miniftry upon himfelf, as that he ac- quired great fame in many countries around, which could not be done in a fhort time, we may believe that the laft verfe of the prophecy allots the firft [ *9<> ] half of the feventieth week (or three years and an half) to the time of Johns preaching ; at the end of which time he baptized Chrift, who was then en- tering into the thirtieth year of his age (according to St. Luke J and then Chrift took his public miniftry upon himfelf for the remaining half of the feventieth week ; at the end of which he was cut ofT by the wicked and felf-hardened Jews, and fo put a virtual end to all their facrifices and offerings; which finally ended with the deftru&ion of their city and temple about 37 years after. So that, in the firft place, taking the whole of the prophecy together, as in ver. 2 5, and then dividing it into four different periods or parts as above men- tioned ; it will very naturally run thus$ From the time of Ezra's receiving the com- 7 w , rnandment to repair Jerufalem, until the> Weeks * Years, expiation of Sin by Christ ^ 70 or 490 For the time of thefe repairs — — - *- 7 or 49 From the finiftiing of thefe repairs to the 7 coming of Christ by his meffenger John > 62 or 434 the Baptift 3 From that time to the end of John's mini- 7 1 * ftry, and the baptifm of Christ — \ 2 or 3i From thence to the end of Christ's mini- 7 , x ~ ftry, by his death on the Crofs J * or 3* ■ ■ .<* In a]l . ..i n .. 70 or 490 [ *9* ] The beginning of thefe feventy weeks of years being found to be in the 457th year before Chrift's birth, at the time of the Jewifh Paflbver, their ending muft have been at the Paflbver in the 3 3d year after the year of his birth : and confequently, according to this pro- phecy, our Saviourwas crucified at the end of 490 years after Ezras commiflion. 'Tis plain from all the four Gofpels, that the crucifixion was on a Friday ; becaule it was on the day next before the Jewifh fabbath ; and as above men- tioned, on the day the Paflbver was to be eaten (at leaft) by many of the Jews. The Jewifh year con filled of twelve months, as meafuredby the Moon, which contains 354 days ; to which they either added 11 days every year, in order to make their years keep pace with the Sun's courfeof 365 days ; or 30 days in three years. So that, although their months were Lunar, their years were Solar. And they always celebrated the Paflbver on the fourteenth day of the firft Lunar month, reckoning from the 2 firft [ *9* ] firft time of their feeing the New Moon ; which, efpecially at that time of the year, might be when fhe was about 24 hours old ; and confequently their four- teenth day of the month fell upon the day of Full Moon ; and, according to Jofephus, they always kept the Paffover at the time of the Full Moon next after the vernal equinox. But the Full Moon day on which our Saviour was crucified fell on Friday. And as 1 2 Lunar months want 1 1 days of 12 Solar months, the Paflbver Full Moons (as well as all others) fall 1 1 days back every year ; which being more than a week, by four days, makes it, that, in a few neighbouring years, there can- not be two Paffover Full Moons on the fame day of the week. And when this anticipation would have made the Paffo- ver Full Moon fall before the equi- noctial day, they fet it a whole month forward, to have it at the firft Full Moon after the vernal Equinox ; which puts it off the fame day of the week again. The [ 1 93 ] The difpute among chronologers, about the year of our Saviour's cruci- fixion, is limited within four or five years at moft. And it certainly was in the year in which the Paffover Full Moon fell on a Friday. And- I find, by calculation, that the only Paffover Full Moon which fell on a Friday, from the 20th year after our Saviour's birth to the 40th, was in the 4746th year of the Julian period; which was the 33d year of his age, reckoning from the beginning of the year next after that of his birth, accor- ding to the vulgar Mr a thereof: and the faid Paffover Full Moon was on the third day of April. And thus we have an aftronomical demonftration of the truth of this an- cient prophecy, feeing that the prophe- tic year of the Mefliah's being cut off was the very fame with the aftrono- mical. Befides, we have the teftimony of a heathen author, which agrees with the fame year. For Phlegon informs us, C c that [ 194 ] that in the fourth year of the 2020! Olympiad (which was the 4746th year of the Julian period, and the 33d year after the year of Chrift's birth) there was the greater!: eclipfe of the Sun that ever was known ; for the darknefs lafted three hours in the middle of the day : which could be no other than the darknefs on the Crucifixion-day ; as the Sun never was totally hid above four minutes of time, from any part of the Earth, by the interpolation of. the Moon* If P^/^^hadbeenanaftronomer, he would have known that the faid dark- nefs could not have been occafioned by any regular eclipfe of the Sun ; as the Moon was then in the oppofite fide of the heavens, on account of her being Full. And as there is no other body than the Moon that ever comes be- tween the Sun and the Earth, it is evi- dent that the darknefs at the crucifixion, was- miraculous, being quite out of the ordinary courfe of nature. There [ «95 3 There have been great difficulties about our Saviour's eating the Pafchal lamb on the evening of the day before it was eaten by the Jews. But I ap- prehend this difficulty may be eafily removed, when we confider that the Jews began their day in the evening, and ended it in the next following evening. So that, although it was on a different day, according to our way of reckoning, it was ftill the fame day ac- cording to theirs. A nd we do not find that they brought in his eating the lamb on the Thurfday evening as any accu- fation againft him : which they would undoubtedly have been glad to do, if they could have made a handle of it for that purpofe. C c 2 A Table m [ 196 ] 1? Old Stile. Years le:"s than an Handled. • 1° -Si o 28 56 84 I 29 57 8? 2 3 C •58 8f, ■5 J 31 m 87 4 32 60 8 Mr 33 • 89 6 34 9 c 35 63 9' b 36 it 9 z y *7 93 IC 58 66 94 I ! 39 °7 9" 12 4 C 68 96 13 H i ^ 71 99 72 i r 45 73 1 8 46 74 19 47 75 20 48 76 21 49 77 .'2 5° 78 23 5 1 79 -'4 52 80 2 5 53 81 26 54 82 2 7 5< 83 Hundreds of Years. 0 100 200 3 CO 400 500 60c 700 800 900 1000 1 100 1 200 1 30c 1400 1500 1600 1700 1800 1900 2COC 2100 2200 2300 2400 2?00 2 too 27OC 2800 2900 3000 3100 320O 3300 3400 3 5°° 3600 3700 3800 39OO 4000 4IOO 4200 4300 4400 4500 4(300 4700 480C 4900 5000 5 100 5200 53° 5400 55°° D C C B B A A G G F F E E D E D C B A G F F E D C B A G G F E D C B A B A A G G F F E E D D C C B C B A G F E D D C B A G F E E D C B A G F G F F E E D D C C B B A A G A G F E D C B B A G F E D C C B A G F E D E D D C C B B A A G G F F E F E D C B A G G F E D C B A A G F E D C B C B 3 A A G G F F E E D D C D C B A G F E r. JJ t> a A A r> ij r F E D C B A G A G G F F E E D D C C B B A B A G F E D C C B A G F E D D C B A G F E F E E D D C C B B A A G G F G F E D C B A A G F E D C B B A G F E D C C W ] Old Stile. -*4 fa ^5 Years left than an Hundred. o 28 5 6 84 1 29 57 85 2 3° 58 8 b 3 3 1 59 87 4 3 2 60 88 5 33 61 89 6 34 62 90 7 35 63 9 1 8 36 64 92 9 37 65 93 io 38 66 94 1 1 39 67 9 l I 2 40 68 96 ! 3 4i 69 97 14 42 70 98 '5 43 7 1 99 16 44 7? J 7 45 73 18 46 74 '9 47 75 20 48 76 2 1 49 77 22 5° 78 2 .3 5 1 79 2 4 5 2 80 25 53 81 26 54 82 27 55 83 Hundreds of Years. 0 1 00 200 300 400 500 600 700 8b c goo 1 000 1 100 1200 1300 1400 1500 u6oo 1700 1800 1900 2000 2 IOC 2200 23QO 2400 2500 2600 2700 28OO 2900 30.QO 3100 3200 33°° 3400 35QO 3600 37QO 3800 3900 4000 4100 4800 42OO 43C? 4400 4500 4600 4700 49OO 5000 5 100 5200 '5300 5400 5500 D C E D F E G F A G B A C B B C D E F G A A B C D E F G G, A B C D E F F E G F A G B A C B D C E D D E F G A B C C D E F G A : B B C D E F G A A G B A C B D C E D F E G F F G A B C D E E F G A B C D D E F G A B C C B D C E D F E G F A G B A A B C D E F G G A B C D E F F G A B C D E E D F E G F A G B A C B D C C D E F G A B B C D E F G A A B C D E F G G F A G B A C B D C E D F E E F G A B C D D E F G A B C C D E F G A B B A C B D C E D F E G F A G G A B C D E F F G A B C D E E F G A B C D [ i 9 8 ] Dominical Letters for fa New Stile. 1752 R A 1787 *753 a VJ 1788 i7?4 p 1 / oy mi 1 j 1 /yu 175& 1757 T 702 1758 1701 '759 1760 F E T 70 c 1 76 1 I 7 06 1 762 I 7f)7 1 /y/ *7 6 3 "D J5 1 ""98 I7O4 a g 1 7no I 7 6 5 P ,800 I yUU 1801 1767 xJ 1802 1700 D ,Q n , I 0 OJ 1769 A 1804 l80S 177O G 1771 F 1806 1772 E D 1807 1/73 C l8o8 1774 B 1809 1775 A 1S10 17/6 G F 181 1 1777 E 1812 I778 D 1813 1779 C 1814 I780 B A 1815 I781 G 1816 1782 i7 8 3 F 1817 E 1818 1784 D C 1819 1785 B 1820 1786 A 1821 G F E D C B A G F E D C B A G F E D C B A G F E D C B A G F E D C B A G F E D C B A G A Table fbevoitig the Days of the Months for e-ver, both in the Old and New Stile, by the Dominical Letters. Janu. 3i Oftob. 3* Feb. 28 Mar. 31 Nov. 30 April 3° 3 1 Anguft 3 1 Sept. 3° Dec. May 3 1 June 30 A B C D B F G j 2 „ -i 6 7 8 9 10 1 1 1 2 1 4. I c .1 16 l 7 18 1 0 20 2 I 22 25 j 2/1 26 j - •■/ 28 -y 3 W 2 1 I 2 a H- 5 6 / 9 1 0 1 1 1 2 13 14 16 1/ 18 I n 20 2 1 z 3 z s 26 27 zS 20 30 j r I 8 a 5 6 — / 0 IC I 1 1 2 I 3 I 4 T I r j 1 6 / 18 1 Q - y 20 2 1 22 25 P 2 C J 2 b 2 7 / 28 2Q 7 7 O 3 1 I 2, ,1 T . 3 G Q y 1 0 1 ; I 2 1 1 I C 1 5 16 1 7 18 ! 9 zo 2 i 2 2 2 - -■1 * r 2b 27 / 28 20 30 3 I J I c J , 6 / s ci I O I 1 I 2 1 2 1 D 1 z) 1 1 c 16 J 7 it 19 ZO 2 I 22 2 3 24 z 5 16 27 28 29 30 5i ] < l 2 3 4 5 6 7 9 10 1 1 1 2 1 ? H 1 5 rp «7 [8 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 1? 5 1 1 2 3 4 5 6 7 8 9 IC 1 1 12 '3 M- '5 [6 18 [C so 21 22 2 3 24 2q :( V 28 29 30 [ 199 ] By the preceding Tables (p, 196, I 97> 1 9^) tne day °f tne moatn an ~ fwering to any given day of the week, and the day of the week anfwering to any given day of the month, may be found, in the Old Stile, within the li- mits of 5500 years before the year of ChrirYs birth, and 5500 years after it: and, in the New Stile, from A. D. 1752, to 1821 inclufive, as follows: j. For any given year before Chrift, look for the complete hundreds of that year (when its number amounts to hun- dreds) at the head of the Table on page 196, and for the years below or lefs than an hundred, to make up the number of the given year, at the left hand ; and where the columns meet, you have the Dominical letter for the given year. Thus, fuppofe the Dominical letter was required for the 585th year before the year of Chrift 1, which was the 584th before the year of his birth. Under 500 at the head of the Table, and againft 84 at the left hand, I find FE, which is the Dominical letter required ; and t 26b J and (hews the faid year to have been a Leap year ; as every Leap year has two Dominical letters, the firft of which ferves for January and February, and the laft for all the reft of the year. The Dominical letter for any given year after the birth of Chrift is found in the fame way by the Table in page 197. Thus, fuppofe it was required for the year 1747; I look for 1700 at the head of the Table, and downward thence, in that column againft 47 at the left hand, I find D ; which fhews that D was the Dominical letter for the year 1747. Thefe two Tables fbew the Dominical for the Old Stile : and the Table on page 198 (hews it for the New Stile, from A. D. 1752 to A. D. 182 1, 2. Having found the Dominical letter for the given year, look for that letter at the top of the Table fhewing the days of the months (page 198) and under the faid letter, you have all the days of the months which are Sundays in that year, in the divilions of the months. Under [ 201 ] Under the next letter toward the right hand, all the days in the column are Mondays; thofe under the next are Tuefdays ; and fo on. When you are out at tfie right hand of the Table, go back to^the left, and fo reckon on ac- cording to the order of the days of the week. Thus, fuppofe for the 585th year before Chrift, for which the Dominical letter (or letters) was FE ; the firfi: ferving for January and February, and the laft for all the reft of the year ; in the Table, pag. 198, I find, under F, the 6th, 13th, 20th and 27th of Ja- nuary ; and the 3d, 10th, 17th and 24th of February; and then, under E, I find the 2d, 9th, 16th, 23d and 30th, of March and November; the 5th, 1 2th, 19th and 26th of Oclober; the 6th, 13th, 20th and 27th of April and July; the 3d, 10th, 17th, 24th and 31ft of Auguft; the 7th, 14th, 21ft and 28 th of September and December; the 4th, 1 ith, 1 8th and 25th of May; and the ift, 8th, 15th, 22dand 29th D d of / [ 202 ] of June; which being all Sundays iii that year, the reft of the days of the months anfwering to given days of the week, are eafily found. For example; if it was required to know on what day of the week the 28th of May fell, in the abovementioned year ; I look for the 28 th of May in the Table, and I find A ftands at the top of the column in which that day is found : and, as the 25th of May fell on Sunday, 'tis plain that the 28 th of May mull have been on Wednefday. Again, fuppofe it was required to find on what day of the week Chrift mas-day will fall upon in the year 1767, New Stile. The Dominical letter for that year is D. Then, under D in the di- vifion for December, in the Table, I find that the 6th, 13th, 20th and 27th are Sundays ; and confequently, as the 20th of December falls on Sunday, the 25th (or Chriftmas-day) muft be on Friday. More examples would be fu- perfluous. How C 203 ] How to divide circles and ftraight lines y into any given number of equal parts ^ whether odd or even. When the given number of equal parts, into which a circle, or a ftraight line is to be divided, are even, and can be divided by 2, 3, 4, &c. the operation is too eafy to need any defcription : but when the given number of parts is odd, as 365, 59, or 31 (which are numbers often wanted) 'tis found difficult to di- vide them, even by a great many tri- als with the compaffes. In order to avoid this difficulty I fhal! fhew a method, by which it is as eafy to divide either a given circle, or a gi- ven ftraight line, into any odd number of parts, as into any even number ; and have all the fpaces between the divivion- lines as equal among themfelves as is fufficient for the purpofe : provided the operator has a good fedor, knows how to open it till the two 6o's on the line of chords are as far afunder, when tried D d 2 by [ 20 4 ] by the compafles, as is equal to the length of the radius, or femidiameter, of the given circle. There can be no given number of odd divilions or parts, but may have as many fubtradted from it as will reduce it to an even number, which may be bifecled, trife&ed, or quartered, &c. And there- fore, by finding the length of an arc in the circle that will bear the fame pro- portion to the odd number taken off, as the whole circle bears to the whole given number, this arc may be eafily divided into as many equal parts as are contained in the odd number which was fubtracled ; and then, the remain- ing number being even, the remaining part of the circle may be eafily divided into that number. All circles contain 360 degrees. Therefore, as the whole number of parts, into which the circle muft be di- vided, is to 360, fo is the number of parts fubtracled to the number of de- grees, and parts of a degree, contained in the arc in which they muft be di- vided. I 20 5 ] vided. Thus, fuppofe it was required to divide a given circle into 365 equal parts : fubtracl five of thefe parts, and there will remain 360, which may be firft divided into fix equal parts, each of thefe again into fix, and each of thefe laft into ten; by which there will be 360 in all. Now fay, as 365 parts is to 360 degrees (the whole circle) fo is the five parts fubtra&ed to the arc they will fill ; which arc, by the calculation, will be found to be 4 degrees, and 93 hundred parts of a degree ; which is a little more than 9 tenths. Therefore, having taken the length of the femidiameter of the given circle by your compaflfes ; open the feclor fo, as that the two points of the compaffes may reach (crofs-wife on the feclor) from 60 to 60 degrees in the line of the chords ; and keeping the fedor at that opening, take off 4. degrees and 93 hundredth parts of a degree (as near as you can guefs by the eye) crofs-wife, from the line of chords, near their be- ginning at the joint; and fet that ex- tent [ 206 ] tent with your compafles upon the pe- riphery of the circle, making marks with the points, and divide the fpace between the marks into 5 equal parts; and then divide the reft of the circle, firft into fix equal parts, then each of thefe again into fix, and each of thefe laft into ten ; and fo you will have the whole circle divided into 365 equal parts, as was required. Again, fuppofe a given circle was to be divided into 59 equal parts: fubtract 9, and there will remain 50. Then, as 59 parts are to 360 degrees, fo are 9 parts to the meafure of the arc they will contain ; which, by the operation will be found to be 54.91 degrees. Therefore, fet off 54.91 (or 5 4^) de- grees upon the circle, and divide that fpace into o equal parts ; then divide the reft of the circle, firft into 5 equal parts, and then each of thefe parts into 1 o ; and the whole will be divided into 59 equal parts, as was required. As twice 29I m? -S S < Hexaedron. 8.66025 \o £ § o IJcofaedron. 2 °- 6 4573J tH ^ u " IDodecaedron. Or, as 1 is to the fquare of the fide pf either of thefe Platonic bodies, fo are the above numbers in this proportion, to the fuperficial content of the refpec- five Platonic body. [ 221 ] I']. The diameter of a fphere being given, to find the fide of any of the Platonic bodies, that may be either infcribed in the fphere^ or circumfcribed about the fphere, or that is equal to the fphere. As i is to the number in the follow- ing Table, refpeding the thing fought, fo is the diameter of the given fphere to the fide of the Platonic body fought. The diameter of a fphere being unity, the fide of a That may be infcribed in the fphere, is That may be circumfcribed about the fphere, is That is e- qual to the fphere, is Xetraedron Oclaedron Hexaedron Icofaedron Dodecaedron 0.816497 0.707107 0.577350 0.525731 0.356822 2.44948 1.22474 1.00000 0.66158 o.449°3 1. 644 1 7 1.03576 0.88610 0.62153 0.40883 j 8. Tie fide of any of the five Platonic bodies being given-, to find the diameter of a fphere that may be infcribed in that body, or circumfcribed about it, or that is equal to it. As the refpe&ive number in the above Table, under the title, Infcribed, Circum- [ 222 ] Circumfcribedy Equal, is to I ; fo is the fide of the given Platonic body, to the diameter of its infcribed, circumfcribed, or equal fphere, in folidity. 19. The fide of any of the five Platonic bodies being given, to find the fide of either of the Platonic bodies which are equal in folidity to that of the given . body. As the number under the title Equal* againft the given Platonic body, is to the number under the fame title againft the body whofe fide is fought, fo is the fide of the given Platonic body to the fide of the Platonic body fought. JRules for finding the folid contents of Bodies* I. To find the folid contents of a Sphere or Globe, Multiply the diameter of the fphere twice into itfelf (which is cubing it) and [ "3 ] and the product by 0.5236 ; the laft product is the folidity required. 2. To find the folid content of a Spheroid. As 14 is to 11, fo is the fquare of the conjugate diameter, multiplied by two thirds of the tranfverfe diameter, to the folid content required. 3. To find the folid content of a Cube. Multiply the fide of the cube twice into itfelf, and the product will be the folid contents thereof. 4. To find the folid contents of a Paralle- lopipedon. Multiply the length by the breadth, and the product by the depth ; the laft product will be the folid content re- quired. 5. To find the folid contents of a Prifm. Multiply the area of the triangular bafe by the height of the Prifm ; and 3 -the [ 224 ] the product will be the folid contents thereof. 6. To find the folid contents of a Com> and alfc of a Pyramid. As 3 is to the area of the bafe, taken in any meafure, fo is the perpendicular altitude of the Cone, or of the Pyramid, to its folid contents, in the fame mea- fure. 7. To find the folid contents of theFruflum - of a Cone y in cubic meafure* Multiply the diameters at top and bottom into one another, and to their produd add a third part of the fquare of their difference : multiply this fum by 0.7854, and the product: fhall be a mean area ; which being multiplied by the perpendicular height, the lalt produtf: fhall be the folid content, in cubic meafure, of the whole, in fuch parts (as inches, feet, &c.) as the dia- meters and height were taken. 8. To t m 1 8. To find the folidity of the Fruftunt of a Pyramid. This is done in all refpefts as in the Fruftum of a Cone, only having re- fpect to the figures of its flat bafe and top, as they may be triangular^ fquare, hexagonal, 8cc. 9. To find the folid contents of any of the five Platonic bodies. As i is to the cube of the fide of any of thefe bodies, fo is 0.117851 to the folid contents of the Tetraedron, 0.4174.04 to that of the Octaedron, 1. 00000 to that of the Hexaedron, 2. 18 1695 to that of the Icofaedron, and 7.663199 to the folid content of the Dodecaedron. 10. To find the folid contents of any ir- regular body, even if it were a Gcofe- berry bufh y provided you have a veffel that will fully hold it. Let the veffel be filled quite up to the brim with water, and weighed in a G p- balance • f 226 ] balance: then put the irregular body into the veffel, till it be quite covered with the water, and it will caufe as much water to run over, as is equal to its whole bulk. This done, take the body out of the water, and then find how much lefs the veffel weighs, than it did, when full of water, before the body was put into it. Reduce this de- ficiency of weight into Troy grains, and divide the number of grains by 253.18287 (becaufe the weight of a cu- bic inch of common water is 2 5 3 . 1 8 2 8 7 grains) and the quotient will be the iolid contents of the body in cubic in- ches, which may be reduced to cubic feet by dividing the number of inches by 1728, the number of cubic inches in a cubic foot. N. B. The outfide of the veffel muft be wetted when it is full of water, and its weight taken, before the body be put into it ; for otherwife, part of the water which the body caufes to run over, when it is immerfed, will flick [ 227 ] flick to the outride of the veffel, and thereby give a falfe conclufion. ii. To find the [olid contents of a Cylin- der ', in cubic inches. As i is to 0.7854 (or rather to 0.78,5399) fo is the fquare of the dia- meter of the Cylinder, taken in inches, to the number of fquare inches contained in the area of the bafe of the Cylinder: which number being multiplied by the height of the Cylinder, taken in inches, gives the folid content thereof in cubic inches. Now, fuppofing the Cylinder to be hollow, and thefe meafures to be taken in the infide ; we may find how much it will hold, in Ale gallons, Wine gal- lons, Corn gallons, or Corn bumels, thus : Divide the content in cubic inches by 282, and the quotient will be the number of Ale gallons ; by 231, and the quotient will be the content in Wine gallons \ by 268.8, and the quotient Gg 2 will [ 228 ] will be Corn gallons ; and by 21 50,42, and the quotient will be the content in Corn bufhels. j 2, To gauge a common Cajk or BarreL Meafure the infide diameters of the Cafk at the end and middle, and take their difference in inches. Multiply this difference by 0.7, and add the pro- duel to the diameter at the end ; which will give the mean diameter of the cafk, very nearly, as if it were a cylindrical veflel of the fame contents with that of the cafk 2 which contents may be found in Ale gallons, Wine gallons, Corn gallons, or Corn bufhels, by the foregoing Problem. 1 3» To gauge a common V at. Multiply the diameters at the top and bottom (taken in inches) into one another, and to their product add one third part of the fquare of their differ- ence ; then multiply this fum by o . 78 54, [ 229 ] and the product will be a mean area, as if the veffel was cylindrical. Mul- tiply this area by the perpendicular height of the Vat, and the product will give the contents thereof, in the number of cubic inches that it will hold : which number may be reduced into Ale gal- lons, Wine gallons, Corn gallons, or Corn bumels, as above. A Table, by which the quantity and weight of water in a cylindrical pipe of any given diameter of bore, and perpen- dicular height, may be found : and confequently, the power may be known that will be fufficient to raifethe water to the top of the pipe, in any pump, or other hydraulic engine. *T3 m Diameter of the cvlindric bore i Inch. t high. Quantity of water in cubic Inches. Weight of Water in Troy ounces. In Avoirdu- poife ounces. I 2 3 4 5 6 7 8 9 0.42477 8 ! 18.8491; 5623, 28.2743343 37.699I 1 24 47.1238905 56.5486686 6S.97344 6 7 75.3982248 84.^230029 4.9712340 9.9424680 14.9*37020 19.8849360 24.8561700 29.8274040 34.7986380 39.7698720 44.7411060 5-454*539 10.9083078 16.3624617 21.8166156 27.2707695 32.7249234 38.1790773 43.6332312 49.0873851 [ 230 ] For tens of feet high, remove the decimal points one place forward ; for hundreds of feet, two places ; for thou- fands, three places ; and fo on. Then multiply the fums by the fquare of the diameter of the given bore, and the pro- duds will be the anfwer. Example; Qu. Tie quantity and weight of water in a cylindrical pipe 85 feet high, and jo inches diameter. The fquare of 10 is 100. Feet high. Cubic inches. Troy ounces. Avoird. ounces. 80 s 753.982248 47.123890 397.698720 24.856170 436.332312; 27.270769 85 80 1. 1 06 1 38 mult, by 100 422.554890 100 463.603081 100 An/ 801 10.61 3800 & 22 5 5.489000 46360.308100 Which number (80110.6) of cubic inches being divided by 231, the num- ber of cubic inches in a Wine gallon, gives 342.6 for the number of gallons: and the refpective weights (4225 5.489, 2 and [ 23* ] and 46360.3) being divided, the for- mer by 12, and the latter by 16, give 3521.29 for the number of Troy pounds, and 2897.5 for the number of Avoirdupoife pounds, that the water in the pipe weighs. So much power would be required to balance or fupport the water in the pipe, and as much more to work the engine as the fri&ion thereof amounts to. Concerning Pumps. In all Pumps, the prefiure of the co- lumn of water, or its weight felt by the working power, when raifed to any given height above the furface of the well, is in proportion to the height of the column, confidered throughout, as if it were equal in diameter to that part of the bore in which the pifton or bucket works. The advantage or power gained by the handle of the pump is the fame as in the common lever ; that is, as great as the length from the axis of the handle to [ *3* ] to its end where the power is applied^ exceeds the length of the other part of the handle, from the axis on which it turns, to the pump rod wherein it is fixed, for lifting the pifton and water. In the making of pumps, the dia- meter of the bore where the bucket works mould be proportioned to the height which the pump raifes water above the furface of the well, as that a man of ordinary ftrength might work all pumps equally eafy, let their heights be what they will. The annexed Table fhews how this may be done, and what quantities of water may be raifed in a minute by one man, fuppofing the handle of the pump to be a lever in- creafing the power five times. N. B. In the quarto edition of my book of Lectures, pag. 75, laft paragraph, and line 3 of column 1, in page 76, for bucket read furface of the water in the well. A Table [ 2 33 ] Find the given h3 5 65 2.72 12 4 70 2.62 11 5 75 2 -53 10 7 80 2.45 10 2 85 2.38 9 5 90 2.3 1 9 1 95 2.25 8 5 1 100 1 2 19 8 1 weight compared with Avoir dupoife weight, 175 Troy pounds are equal to 144 Avoircupole pounds. 175 Troy ounces are equal to 192 Avoirdupoife ounces. 1 Troy pound contains 5760 grains ; and I Avoirdupoife pound contains 7000 grains. 1 Troy ounce contains 480 grains ; and I Avoirdupoife ounce contains 437.5 g-ains. I Avoirdupoife dram contains 27.34375 grains. 1 Troy pound zz 13 ounces 2.65 1428576 Avoird. drams j and I Avoirdupoife pound is equal to i pound, 2 ounces, II pennyweight, 16 grains Troy. Hh By C 2 34 } By the following Table, we may fimf how much of either of thefe weights is contained in any given number of pounds in the other. Tr. P. i 2 3 4 5 6 7 8 1 9 Avoird. Pounds. Av. P. 1 2 3 4 5 6 7 8 9 Troy pounds. j 0.822857 142857 143 1.64:5714285714286 2.46857 1 42857 1429 3.291428571428572 4.114285714285715 4.937142857142857 5.760000000000000 6.582857142857143 7.40^ 7 1 4285714286 1. 21527777777778 2 430555555 5555 6 3- 645 8 3333333333 4.861 1 1 1 11 1 1 1 1 1 1 6.07638888888889 7.21966666666667 8.50694444444444 972222222222222 , 10.93750000000000 For tens of pounds, remove the de- cimal points one place forward ; for hundreds of pounds, two places ; for thoufand, three places ; for tens of thoufands, four places ; and lb on, as in the following Examples. When any fractions remain in the laft fum, reduce them to the known parts of a pound, by the common me- thod of reducing decimals to the known parts of an integer : remembering, that in Troy weight, 12 ounces make a pound, 20 pennyweight make an ounce. C 2 35 ] ounce, and 24 grains make a penny- weight : and that, in Avoirdupoife weight, 16 ounces make a pound, and 16 drams make an ounce. Example I. In 175 Troy pounds , Qu. How many Avoirdupoife pounds f Avoirdupoife. 82.28571428157143 57.6000000000000 ds/hv. 144. 4.1 142857142857 , 1 44.0000000000000 Example II. In 144 Avoirdupoife pounds Qu. How many Troy pounds f Troy. 121.5277777777778 48.61 1 1 1 1 1 1 1 1 1 1 1 Anfip. 175, 4.861 nil in iii 1 7 5.0000000000000 Troy. lOO ' 70 5 Avo. 100 40 4 M4 H h 2 Example [2 3 6] Example III. Jn 72 Avoir dupoife pounds, Qu. Haw Ayo. 70 72 P. weight f Troy. 85.06944444444444 A/ifwe r , 2 -43°5555S55555 6 *7;5 pounds, 87 pounds 87.50000000000000 6 ounces. P. In common pradice, 'tis fufficient to take out the decimal parts to five or fix figures. " By four weights, viz. 1 pound, 3 pounds^ 9 pounds, a?td 2 j Pounds, to ^ weigh 40 pounds \ or any number of pounds from 1 40, Pounds. J Scale 4. Scale B. •T3 Sc. B. O c Scale A. 1 1 0 21 2 7>3 9 2 3 1 22 2 7>3>i ■ 9 3 3 0 2 3 2 7 3,1 4 *>3 0 24 27 3 I 9 3.1 2 5 2 7»i 3 9 3 25 2 7 I 7 9,1 3 2 7 27 0 8 9 1 28 27,1 0 9 9 0 29 2 7,a 1 IC 9,1 Q 30 z 7'3 0 1 1 9*3 I 31 2 7>3,i 0 12 9' 3 O 3 2 2 7,9 3>J *3 9'3»i O 33 27.9 3 14 27 i>3>9 34 27,9,1 3 *5 2 7 3>9 35 2 7>9 1 *6 27,1 3>9 36 2 7>9 0 17 27 9,1 37 27,9,1 0 18 2 7 9 38 2 7>9'3 1 J 9 27,1 9 39 2 7>9>3 0 20 z 7»3 — 1 0.007 q 7 1 17.18 10 4.. 76 J 7-793 7 ■2 0 to? 7 *3-S 6 5 r j IQ 16.32 6 8.86 11.325 c j 17 o.co 6 6.69 1 1 090 c J 1 1 I r r 10.534 A T - 22.62 r j 3*33 9 oco 4. T 8 2.0c T 17.21 8-344 A. r 10.76 4 I O08 0 _ 0.001 A. 2 20.2 1 4 8.- 1 0,j I 7- 8 3? 3 17 C. C 2 j j A T 2 no 7.320 I IC j 21. 07 I 3.400 I 8 I4. 1 I I O 20 2.710 I 7 5.2O I 7.88 2.579 0 19 I8.43 I 2.03 I,s /3 0 19 5 83 I 0.89 1.823 0 10 20.77 0 9-54 1.029 0 10 20.5 1 0 9.51 1.028 0 10 13-18 0 9- 2 3 1. 000 0 10 1 1.42 0 9.20 •993 0 9 19.73 0 8.62 •93i 0 9 3- 2 7 0 8.02 .866 0 7 1 4.00 0 7.46 .720 0 2 12.77 0 2.21 .240 0 0 0.28 0 0.009 .001 Take away the decimal points from the numbers in the right-hand column, and reckon them to be whole numbers; and they will mew how many Avoir- dupoife ounces are contained in a cubic foot of each of the above bodies in the Table. A Table [ 239 ] ■ A Table of the different V elocities and Forces of the Winds. Velocity of Perpen- j dicular the Wind. force on one foot Area, in Miles — in Feet in pounds one one Avoir- Hour. Second. dupoife. I 2 47 .005 2 2.93 .ozo 3 4.40 .044 4 5.87 .079 5 7-33 .112 IO 14.67 .492 15 22.00 1 .107 20 29-33 I.958 25 36.67 3-°75 4.428 30 44.00 35 5133 6.027 40 58.67 66.00 7.872 45 9.963 5° 73-33 I 2.3OO 60 88.00 I7.7I2 80 11 7-33 3I.488 : 100 146.70 49.2 00 Common apppellations of the forces of Winds. up trees, Not perceptible. Juft perceptible. Gentle pleafant wind. Pleafant brifk Gale. Very brifk. High Winds. Very high. A ftorm or tempeft. A great ftorm. A hurricane. A hurricane that tears and carries buildings, &c. before it. The force of the Wind is as the fquare of its velocity. That the force of the wind is as the fquare of its velocity, I have often proved by experiments made on my Whirling Table, Directions t 240 J Dire&ions for Mill-wrights. To have a water-mill for grinding corn, in the greateft degree of perfec- tion* the float-boards of the undermot water-wheel, or the buckets of the overfhot water-wheel, ought to move with a third part of the velocity where- with the water ads upon them ; and when the wheel goes at that rate, the millftone ought to make about 60 re- volutions in a minute : for, when it makes but absut 45 or 50, it grinds too flow ; and when it makes about 70 or 75, it heats the meal too much. When the wheel turns round with a third part of the velocity of the water^ the water has then the greateft power to turn the mill. On thefe principles I have calculated the following Table, adapted to a water- wheel 1 8 feet diameter; which, I thinly is a good common fize. In the firfl column, find the per- pendicular height of the fall of water, 5 ' m [ 24-1 ] in feet ; then, againft that height, in the fecond column, you have the ve- locity of the water in feet per fecond ; in the third, the velocity of the wheel ; and in the fourth, the number of cogs in the wheel, and ftaves in the trundle, for cauling the millftone to go about 60 times round in a minute. 1 i 72* ins [ 242 ] "The Mill-wright s Table* 1 2 3 4 5 Perpen- dicular Velocity of the Wa- ter per Velocity of the Wheel per 0 0 OTQ 3' a 5 0) Number of Turns of the Millftone height or the Second. Second. 5' Er Fall of Water, in Feet. h-j 0 w — > 0 -*-J &T 0 09 jg ft -1 per Mi- nute, by m n T) >-o O £u O 2 r* c/> n> <1> O ju rrr Pi. c D- * thefeCogs andStaves. i Q O ,02 2 .by * Z7 6 59.92 2 1 I •4° 3 • 0^ I) 08 y° 7 00,00 3 *3 .89 4 .63 8 60.14 4 16 .04 5 •35 y 59.87 • 5 17 •93 5 .98 e 5 9 59-84 c o 19 .64 6 •55 78 1° y s ^ • 00. IO 7 21 .21 7 .07 72 Q 60.OO 8 22 .68 7 .56 67 9 59.67 9 24 .05 8 .02 70 10 59-57 IO 2 S •35 8 •45 67 10 60.0.9 i i 26 •59 8 .86 64 10 6c. 1 6 12 27 •77 9 .26 61 10 59.90 *3 28 .91 9 .64 59 10 60.18 H 3° .CO 10 .00 5 b 10 59-36 15 3 1 .05 10 •35 55 10 60.48 16 3 2 .07 IO' .69 S3 10 60.10 17 33 .06 1 1 .C2 5i 10 59.67 18 34 .02 1 1 •34 5° 10 60.10 l 9 34 •95 1 1 .65 49 10 60.61 20 35 .86 1 1 •95 47 10 59. eg 0/ [ 2 43 ] Of the difference between the apparent. Level and the true. When a plumb line hangs freely, it hangs dire&ly toward the center of" the earth : and a right line, croffing the direction of the plumb line at right an- rrles, and touching the Earth's furface juft below the plummet, is a level at that point of the Earth's furface. But, if this right line be continued from that point, keeping ftill perpendicular to the plumb line, it will rife above the Earth's furface, became the Earth is of a globular fhape : and this rifing will be as the fquare of the diflance to which the faid right line is produced. That is, however much it rifes above the fur- face at one mile's diftance, it will rife four times as much at the diftance of two miles ; nine times as much at the diftance of three miles ; fix teen times as much at the diftance of four miles ; and fo on. And therefore, if two le- vels are taken at two points of the 1 i 2 Earth's [ 244 ] Earth's furface which are at any con- fiderable diftance (as fuppofe a mile) from each other, the level lines pro- duced will interfecl each other at a certain angle : and although either of them, fo produced, will appear to be a true level, yet it can be fo only at that point of the Earth's furface from which it was produced : not at any other. The height to which a level line, pro- duced from any given place, rifes above any other place, is the height of the apparent level above the true at that other place: the quantity of which height is fhewn by the following Table, for all diftances within the length of a degree of a great circle upon the Earth's furface. There is a Table of the fame fort in Dr. Long's Aftronomy, which differs but two inches from this (which I have computed) in the height of the apparent level above the true, for a whole degree, or 60 geographical miles ; which are longer than 60 Englifh miles by 48840 feet. By 4 [ 2 4S ] By the moft accurate meafures of the length of a degree on the Earth's fur- face, the whole 3 60 degrees of the Earth's circumference contain 1 31630400 feet, or 24930 Englifh miles: which in geographical miles (allowing 60 to a degree of a great circle) make only 2160c. So that, a geographical mile contains 6094 feet, which exceeds the length of an Englifh mile by 814. In the Table, a geographical mile (which I have often thought mould be the univerfal ftandard length of a mile) is called a minute, becaufe it is the 60th part of a degree ; and the 60th part of fuch a mile is called a feconcL As the furface of water naturally an- fwers to the curvature of the Earth's furface (fuppofing no hills or eminences thereon) 'tis plain that if a long ftraight channel was made, fo as to have its middle part level at any part of the Earth's furface, and the reft continued out both ways in direction of an ap- parent level from that place ; if water fhould come in at either end of the i channel, [ h6 3 channel, it would run to the middle thereof: and, if the channel was all of an equal depth, the water would run over at the middle before the channel could be filled at both its ends. And confequently, if a diftant fpring appeared by a levelling inftrument to be juft on a level with the houfe, the water might be brought in a ftraight channel from the fpring to the houfe ; or in pipes, if there was an intervening valley ; be- caufe the water will rife in crooked pipes, till its furface at both ends, is equally diftant from the Earth's center. A Table [ 247 1 A Table /hewing the height of the ap- parent level above the true^ to the looodtb part of an Inch. Seconds, i 2 ! X •: 5 6 7 S o *3 '9 20 21 22 23 24 25 26 27 28 29 30 o O-c C 13 Feet. Inches. 203 304 406 507 609 710 8iz 914 1015 1117 1218 1320 1 42 1 1523 1625 1726 1828 1929 2031 2132 2234 2336 2437 2539 2640 2742 2843 2945 3047 6.8 1.6 8.4 3- 2 10.0 4.8 11.6 6.4 1.2 8.0 2.8 9 .6 4.4 11. 2 6.0 0.8 7.6 2.4 9.2 4.0 10.8 5.6 0.4 7.2 2.0 8.8 3 .6 10.4 5.2 0.0 0.027 0.048 o 0.074 — 0.106 a. bQ [ 248 ] The Table continued* Seconds. Feet. Inches. ... Inches. 3 1 3148 6.8 z.039 3 2 0 3 2 5° 1.6 3 .046 © 33 CO 335 1 8.4 ■3 218 "Z 34 c 3 3453 3-2 3.410 2 35 O g 3554 10.0 1 n In 3656 4.8 —j 3.OZ9 u 37 0 cd 0 5.212 « 43 u •5 4367 4.4 .£> ea ■S 44 e 4468 11.2 "33 5.720 S 4> 0 Ol, 457° 6.0 > c 082 v9° 3 £ 46 s 4672 - O.8 6.25 2 « 47 "0 4773 7.6 J- 6.527 £ 48 "0 4 8 75 2.4 s. Ot V. ouo 0 49 4976 9 2 at 7.094 JS 50 4-4 5078 4.0 V •5 7 "9.1 7.307 5 1 5*79 10.8 7.68c 0 cz «.> H 5281 5.6 0 *7 r\Qr\ 7.909 2 53 n3 5383 0.4 8.3OO <§ 54 5484 7.2 CU M 8.616 =5 55 « 5586 2.0 7 U 4265 8 '% 43 5.209 8 48752 4J 3 5* 8.763 ^3 9 0 127 7 716 U I 2 1 5 7Q222 ca 149 10.025 l 4 c U m 8.836 g I f 0 9I4IO u 199 5-307 o <-£ 16 3 Q 7 COi. a 226 1 1.052 rt-! I 7 0 103598 QJ 256 2.070 QJ •*» ►Q 18 109692 Oh 287 2.362 O *9 «l 1 1 5786 p t eg 3'9 1 1.928 QJ 20 u 121800 CJ -G 354 6.768 21 ca 127974 Uh 39° 10.882 o 22 .g 134068 O 429 0.269 CJ C 2 3 -a 140162 JS 468 10.931 m 24 146256 510 6.866 25 CO 195b 0.956 Oh Cl, 2042 3'464 rs u Z I 2 0 3- 2 45 _<■» 22l6 0 inn 2305 6.629 O 2396 IO.232 !> 2489 1 1. 108 2584 9.259 ft) 268l 4.683 2 779 9.381 2879 "•353 2981 10.598 3085 7.199 3191 0.912 By [ 2Si ] By the preceding Table, the length of an arc, in feet and inches, on the Earth's furface, may be found ; if its meafure be known in minutes and fe- conds of a degree. Thus, fuppofe the length of the arc be 10 feconds, which is the fixth part of a geographical mile ; its meafure is 1015 feet 8 inches: and an arc of one minute of a degree, which is one geographical mile, is found to be 6094 feet. We may alfo find how far one can fee in a true horizon at fea, when the eye is raifed to any given height above the furface of the water. Thus, fuppofe the eye of an obferver on a {hip at fea to be 22 feet, 2 inches, above the fur- face of the fea, he will fee to the di- ftance of 30470 feet all around him ; or to the diftance of 5 geographical miles : for againft 22 feet, r.923 (which may be efteemed 2) inches, in the right hand column, is 30470 feet, in the middle column ; and 5 minutes or geo- graphical miles in the firft. K k 2 Again, [ 2 S2 ] Again, fuppofing that a gentleman was to make a canal in a piece of ground which is half a geographical mile long, and appeared to be truly level by the common levelling inftrument ; this being 30 feconds of a degree, or 3047 feet, in length, the height of the appa- rent level above the true, at that di- ftance, is 2.659 inches: and to much muft the farther end of the ground be funk, in order to have the water equally near the furface of the ground at both ends of the canal. Once more ; fuppofe an obferver to have his eye clofe at the furface of the fea, and that he then juft fees the top of a mountain in the fea, whofe diftance he knows to be juft 60 geographical miles, or 365640 feet; the perpen- dicular height of that mountain, above the furface of the fea, is 3191 feet and T | parts of an inch, If the diftance of the mountain fo feen be more than 60 geographical miles, which is without the reach of the [ 253 ] the Table ; yet its height may be found by the Table, in the following manner. Suppofe the diftance of the mountain to be 90 geographical miles, or a degree and an half, in a great circle upon the Earth: take half the given number of miles or minutes, which is 45 ; and multiply the height of the apparent level above the true, at that diftance, by 4 : and the product will give the perpendicular height of the mountain* Thus, againft 45 minutes you have the height 1794 feet, 1 1.763 inches; which multiplied by 4, gives 7917 feet, 11.052 inches, for the height of the mountain above the furface of the fea* According to the meafures in the Ta- ble, a degree of a great circle upon the Earth contains 6g~ Englifh miles. Of the Mechanical Powers^ ajid of Fri&io?i. From the moftfimple machine to the mo ft compound Engine^ the power or * ■ advantage [ 2 54 ] advantage gained is always as much as the velocity of the moving power ex- ceeds the velocity of the weight or re- finance that is moved ; making proper allowance for the friction of the machine or engine. So that, if the working power moves through a fpace of ten, or an hundred, or a thoufand inches, whilfl the weight or refiftance moves only through the fpace of one inch ; the perfon who works the machine or engine (fuppofing it to have no friction) could raife ten, or an hundred, or a thoufand times as much weight as he could do by his natural ftrength without it. But the time that is loft will be al- ways as much as the power that is gained. The 11 m pie machines by which pow- er is gained, are fix in number,- viz. the Lever ) the Wheel and Axle^ the PullieSy the Inclined Plane^ the Wedge ^ and the Screw. Of thefe fix fimple machines, all the moft compound en- gines are made : for we know of no other fimple machines by which power can be gained. l A lever [ 2^5 ] 1 . A ler/er is a bar laid over any prop that will fupport the weight upon it. If the prop be under the middle of the lever no advantage is gained by it ; for as faft as a man pufhes down one end, the power rifes on the other. To gain power by it, the length of the part or arm between the man and the prop muft be greater than the length of the part or arm between the prop and the weight. And then, as much power will be gained as the length of the longer arm exceeds the length of the fhorter. 2. If an axle turns upon its gudg- eons, and is fixt into a wheel ; and if a rope that raifes the weight coils round the axle, whilft a man pulls a rope that was put round the wheel ; the power gained will be as much as the diameter of the wheel, added to the diameter of the rope, exceeds the diameter of the axle added to the diameter of the rope that coils round it. 3. In the pullies, the power gained is equal to twice the number of pullies in the lower block, to which the weight 7 is [ 2 5 6 ] is fufpended. So that the power gained is always in proportion to the number of parts of the rope by which the lower block and weight are fufpended. 4. In the inclined plane, or half wedge, the power gained is as much as the length of the machine exceeds its thicknefs at the back, on which the ftroke is given by the fledge or mallet that drives the machine. 5. In the wedge, the power gained is as much as the length of both the fides of the wedge, taken together, exceeds the thicknefs of its back, on which the blow is ftruck by the hammer or mallet, 6. In the fcrew, the power gained is as much as the circumference of the circle defcribed by the working power, that turns the fcrew, exceeds the dif- tance between the threads or fpirals of the fcrew. In the lever, the friclion is nothing. In the wheel and axle it is as fmall as the diameter of the gudgeons (added to the power required to bend the rope) is lefs than the diameter of the wheel ; but [ 2 57 ] but it increafes according to the weight with which the axle is charged. The like might be faid of the pullies, if they did not rub againft one another, or againft the fides of the mortifes in the block where they are placed. A new rope of one inch diameter, going over a pulley 3 inches diameter, and pulled with a force equal to 5 pounds, requires a force of 1 pound to bend it ; and a rope two inches diameter requires four times as much force. In the inclined plane wedge, and fcrew, the friction is at leaft equal to the power, becaufe they will fuftain the weight in any po- rtion when the power is taken off. Wood greafed or metal oiled, have nearly the fame friction ; and the fmoo- ther they are their friction is the lefs. Yet metals may be fo highly polimed, as to have their fri&ion increafed by the cohefion of their parts. Wood Aides eafier upon the ground in wet weather than in dry ; and eafier than an equal weight of iron in dry weather : but iron Aides eafier than L 1 wood [ 255 > wood in wet weather. Iron or fteel running in brafs has the leaft fricHan of any. Lead makes a great deal of re- finance. In wood acting upon wood,, greafe makes the motion at leaft twice as eafy. Wheel naves greafed or tarred go four times as eafy as when wet. Smooth foft wood, moving upon, fmooth foft wood, has a friction equal to about a third part of the weight. In, rough wood, the friction is almoft equal to half the weight. In foft wood upon hard, or hard upon foft, the friction is equal to about a fifth part of the weight. in polimed fteel moving upon po- lifhed fteel or pewter, the friction is about a fourth part of the weight : on copper a fifth part, and on brafs a fixth part of the weight, Metals of the fame fort have more friction than different forts. In general, the friction increafes in the fame proportion with the weight. The friction is alfo greater with a great- er [ 259 ] er velocity ; but not fo great, in pro- portion, as the increafe of velocity. To have the friction of machines as little as poffible, they ought to be made of the feweft and fimpleft parts. The diameters of the wheels and pullies ought to be large, and the gudgeons of the axles as fmall as can be confident with their required ftrength, The fides of the pullies ought not to be all over flat, but to have a fmall rifmgin the middle, to keep them from rubbing againft each other's fides, and againft the fides of their mortifes, at a diftance from their axles. Ail the cords and ropes ought to be as pliant as poffible ; and for that end, rubbed with greafe. The teeth of the wheels mould juft fit and fill the openings, fo as neither to be fqueezed nor (hake therein. All the parts which work into or upon one another ought to be fmooth, the gudgeons ought juft to fill their holes, and the working parts muft be greafed. The rounds or fbves of the trundles may be made to turn about upon iron fpindles fixt in the L 1 2 round [ 26o ] round end boards, which will take off a great deal of fridlion. Let the ftrength of all the parts be in proportion to the ftrefs they are to bear; fo as they may laft equally well. He is by no means a perfeel mechanic who does not only adjuft the ftrength to the ftrefs, but alfo contrive all the parts to laft fo as that one fhall not fail before another. When any motion is to be long con- tinued, contrive the machine fo, as that the working power may always move or adt one way, if it can be done. For this is better and eafier performed than when the motion is interrupted, by the power's being forced to move firft one way and then another ; becaufe every new change of motion requires a new additional force to effecT: it, and a body in motion cannot fuddenly receive a contrary motion without great violence, and danger of tearing the machine to pieces. But, when the nature of the thing requires that a motion mould be fuddenly communicated to a body, . or fuddenly [ 26t ] fuddenly ftopt ; let the force act againft fome fpring, to prevent the machine's being damaged by a fudden jolt. When a machine is moved by two handles or winches on the ends of an axle ; the handles are fo placed, as that when one is up the other is down; which is the worn; way poffible of pla- cing them, fave that of their being both up or down together. For, when a man raifes a weight by means of turning a winch, he lofes half his force when the winch is upward, becaufe he pufhes himfelf as much backward as he pufhes the winch forward ; and when the handle of the winch is down, directly below the axle, he lofes half his force, becaufe the winch pulls him as much toward it as he pulls it toward him : and there- fore, the greajteft effect of his force on the machine is when he either pulls the winch upward, on the fide of the axle next to him, or pufhes it downward on the fide fart heft from him : yet, even in thefe cafes, the pulling force is ftron- ger than the pufhing. In [ 262 ] In order to remedy this defect, 3s much as poffible ; the handles fhould be fo placed, as to ftand at right angles to one another : and then, when there is a man at each handle, the effect of the one man's force will be greateft when the effect of the other man's is leaft, upon the machine. Whereas, in the common way of placing thefe han- dles, when the effect of one man's force is the greateft, the other man's is fo too ; and when the effect of that man's force is the leaft, fo alfo is the other's ; which is working at the greateft difadvantage poffible. A mechanical way of laying down the Suns Declinatio?i right, again/1 the days of the months y either on a circu- lar or re&iliiieal IS c ale. The Sun's declination is ufeful on many accounts ; and remarkably fo in thofe kinds of Sun-dials which are fo conftructed, as that they may be fet 5 true, E 263 ] true, in any place where the Sun mines, without the help of a meridian line ; becaufe, when they are truly level, and have the Sun's declination laid down on their ftiles, the dial itfelf being a circular plate placed on the middle of the ftile, and at right angles thereto ; if the dial be turned about till the mad aw of the circular plate touches the Sun's decli- nation for the given day, the dial will then be truly placed, and the ftile (which will be in the plane of the meridian) will caft a friadow on the true folar time of the day. This is the cafe in M. Pardies univerfal dial, which is one of the beffc 1 know of; and which the reader will find particularly defcribed in my Mechanical Lectures, fold by Mr. (Jadelly Bookfeller in the Strand, London. There are Tables near the beginning pf this book which fhew the quantity of the Sun's declination at the Noon of every day of the fecond year after Leap year, which is the neareft mean of all the four years. But, as the declination very feldom comes to integral degrees at [ 264 ] at Noon, it is difficult by thefe Tables to know at what time of the day the declination will amount to compleat degrees without fra&ions ; and confe- quently it is difficult to lay down the whole degrees thereof againft the pro- per times of the days of the year, in a fcale of months ; although every day in the fcale fhould be divided into four equal parts, each whereof contains 6 hours. To avoid this difficulty, I have cal- culated the following Table, for fhewing the times, to the neareft hour of the day, when the Sun's declination amounts to compleat degrees without fractions. Thus, fuppofing it was required to find on what days of the year, and at what hours of thefe days, the Sun's declina- tion was juft 9 degrees ? Look for 9 in the declination columns, and againft it you will find April 12, at 16 hours (reckoned forward from the Noon of the day) Auguft 30th, at o hours (or at Noon) October 1 6th at 3 hours paft Noon, [ 26s ] Noon, and Feb. 25th at 1 hour part Noon. Now, if the 365 days of the year be laid down on a fcale, and each day be divided into 4- equal parts thereon, in fhorter ftrokes than thofe which mark the Noons of the integral days ; each fubdivifion by the fhorter ftrokes will reprefent 6 hours, and any one may truft to the accuracy of his eye in placing the diviftons for the whole de- grees of declination at or between thefe mbdivifions of the days, as they are (hewn by the Table to be at o hours (or Noon) 6 hours, 12, or 18 hours after Noon ; or at any time fooner or later. M m A 'Table [ 266 ) A 'Table Jhewing at what times the Suns declination is whole degrees^ and bis Place in the Ecliptic at thoje times. Dec! Worth De- clination Sun'. i Place Dec Worth De- clination Sun's Place increafes in in me Ecliptic. 2; decreafes in in Eel the ptic. Deg. Mon.D.H. Deg. Mon. D.H. S. ° ' o Mar. 20 5 0 0 X a June 21 5 25 0 0 i 22 T 0 Io 2 33 23 J u] y 3 1 11 19 2 25 7 5 2 22 12 2 19 56 3 27 20 7 33 21 18 8 25 5 6 4 3° 10 ?o 5 20 23 12 a 0 52 5 Apr. 2 1 1 12 39 19 28 2 5 13 6 4 J4 15 1 2 18 Aug. 1 4 9 9 7 7 7 17 49 17 5 0 12 48 8 9 23 20 27 16 8 13 16 14 9 ?? 16 2 3 7 I? II 23 19 29 10 15 11 25 5° H 15 5 22 37 1 1 18 7 28 37 13 ?8 8 ?5 38 I 2 21 6 « 1 27 12 21 9 28 33 13 24 7 4 22 I I 2+ 7 I 2 3 14 27 8 7 23 IO 27 4 4 10 '5 3° 13 10 If 9 3° 0 6 53 16 May 3 2 3 13 46 8 Sept. 2 8 9 33 17 7 1 1 17 12 7 4 ii }2 II >8 M 7 20 51 6 7 '4 14 48 J 9 15 9 24 47 5 9 18 17 21 ZO ?9 20 29 8 4 1 2 9 *9 55 2J 25 1 4 4 3 15 0 22 27 22 / 10 4 2 *7 13 24 58 ?3 |une 9 7 18 41 1 20 3 ?7 27 21 5 55 0 0 0 22 17 0 0 [ 2<$7 ] "The Table concluded. Decl. S. . Souith De- clination increafes in Sun's Place in the Ecliptic. Decl. S. South De- clination decreafes in Sun's Place in the Ecliptic. Deg. Mom .D.H. S. Deg. Mon. D.H. S. - 0 ■ ■'• 1 1 '- o Sept. 22 0 0 Dec 21 9 O O I 2 S 9 2 33 2 3 Jan. 1 6 II 19 2 27 20 5 2 22 9 16 19 56 3 Oft.. 30 9 7 33 21 15 14 25 56 4 2 23 10 5 20 20 10 O 52 5 5 13 12 39 '9 2 4 16 5 »3 6 8 4 15 12 18 Feb. 28 »3 9 9 7 10 18 17 49 17 1 4 12 48 Q O 13 1 1 20 27 10 4 13 16 14 9 16 3 2 3 ,7 15 7 18 19 29 IO 18 21 25 50 H 10 '9 22 37. 1 1 2r 16 28 37 13 13 20 2 5 38 12 24 11 m 1 27 12 16 17 28 33 13 27 9 4 22 1 1 19 13 H 1 33 »4 30 IO 7 23 10 22 8 4 10 15 Nov. 2 13 10 31 9 2 5 1 6 53 16 5 *9 13 46 8 Mar 27 16 9 33 17 9 5 17 12 7 2 7 12 11 18 12 20 20 51 6 4 22 14 48 19 16 17 24 47 5 7 12 17 21 20 21 1 29 8 4 10 2 *9 55 21 Dec, 2 5 2 1 4 4 3 12 H 22 27 22 1 19 10 4 2 15 4 24 58 23 10 6 18 41 1 »7 *5 27 27 1 a 2 1 9 Y? 0 0 0 20 5 0 0 M m 2 A Table [ 263 ] A Table fhewing the Latitudes and Lon~ gitudes of a great maity remarkable Places \ and what the times are at London when it is Noon at thofe places, N. fignifies North Latitude^ S. South Latitude ; E. Eajl Longitude { W. W 7 ,Jl Longitude^ from the meridian of London: F. Forenoon^ and A. Af- ternoon^ at London. Noon at Aberdeen Abo Adrianople Aleppo Algiers Amfterdam Annapolis Royal Archangel Aftracan Azoph Bagdat Barcelona Bafil Batavia Bencoolen Berlin Bern Bologna Bombay Bolton Bridge Town Briftol Bruflels Buda Buenos Ayres Cadiz Cairo (grand) T ip t\ tnn p J_J nn iidinuurgn 1 .04. J-JliDUIl 1 . 1 1 1 1 1.0989 Norimberg 0.78 Naples 0.8928 Pans I.C989 Prague 08 Placentia 1.0204 Rochelle °-73 Rome o-9345 Rouen 1.0989 Seville 0.862 Tholoufe 0.75 Turin °-9433 Venice 1.07 Vienna c.7 1.0865 J-I3S 075 1. 1363 0.71 1. 123.5 1.2048 0.72 0.8928 0.7874 1. 1089 0.9259 0.8928 0.82 1.06 1.23 A "Table for comparing the Englijh Foot with Foreign meafures> in EngliJIj Inches* Englifh foot Amflerdam Paris Rheinland Scotch Dantzic Swedifh Brufiels Lyons Bononian Milan foot Roman palm Naples palm, Inches. 12.000 1 1. 172 12.788 12.362 12.061 1 1.297 1 1.692 10.828 I3-45 8 14.938 15.631 8 -779 10.384 Englifh yard Englim ell Scotch ell Paris aune Lyons aune Geneva aune Amllerdam ell Danifh ell Swedilh ell Norway ell Seville vara Madrid vara Portugal vara Inches. 36.000 45.000 45.000 46.786 46.570 44.760 26.800 24.930 23.380 24.510 33-*27 39.166 44.031 Antwerp C 2 75 ] Antwerp ell 27.170 Bruffels ell 27.260 Bruges ell 2 7-5 5° Bononian brace 25.200 Romifh brace 30.730 Florence brace 22.910 Portugal cavedo 27.354 Old Roman foot 11.632 Perfian arifh 38.364 The fhort pike ) ofConftantinopIe J 2 5»57° The long pike 27.920 The Weight andV due of Gold and Silver Coins. A Troy pound of gold is worth 46 pounds 14 {hillings and 6 pence : for 44 guineas and an half are coined from each pound at the Mint. A Troy ounce is worth 3 pounds 17 fhillings and 10 pence 2 farthings; and a grain is worth 1 penny 3-^ far- things, in coinage ftandard. A Troy pound of Silver (coinage ftandard) is worth 3 pounds 2 fhillings; an ounce is worth 5 fhillings and 2 pence; a pennyweight is worth 3 pence and ^th parts of a farthing ; and a grain is worth about half a farthing. A Five- moidore piece weighs 1 ounce, 14 pennyweights, 15 grains. A 3 pound 12 fh. piece weighs 18 penny- weight, 1 2 grains. A Guinea 5 pen- nyweight, 9^ grains. A Moidore 6 N n 2 penny- C 276 ] pennyweight, 22 grains; and a Piftole 4 pennyweight, 8 grains. The proportion of Alloy in coinage. The ftandard of fterling filver is 1 1 ounces 2 pennyweight of pure lilyer, and 1 8 pennyweights of copper. The ftandard of fterling gold is 11 Troy ounces of pure gold aqd 1 ounce of copper for alloy. Our gold is of equal finenefs with the Spanifh, French, and Flemifh ; but our filver coin has lefs alloy in it than either French or Dutch. Jewijh weights reduced to Englifh. Troy weight. A Shekel, 9 pennyweight, 2.57 grains. An hundred Shekels, or 3 lb. 9 oz. 10 p. w. 17 grains, make a Manch : and 50 Manches, or 109 lb. 8 oz. 15 p, w. 10 gr. make a Talent. jfewijh [ 277 ] Jewijh Dry-tneafure reduced to Englifi Corn-tneafure, A Cab, 2 1 pints. An Omer, 5 h pints, A Seah, 1 peck 1 pint. An Ephah, 3 pecks 3 pints. A Lethech, 1 6 pecks \ and an Homer Choron 32. "Jewijh Liquid -meafure reduced to Engli/h. A Log, 3 quarters of a pint. A Cab, 3 pints. A Hin, 1 gallon 2 pints. A Seah, 2 gal]. 4 pints. A Bath or Ephah, 7 gallons 4 pints. A Coron, or Homer, 75 gallons 5 pints ; all in wine meafure. 'fewijh money reduced to Englijh. 7' A Gerah, 1.36^. A Bekah, 1 jh. 1.7 d A Shekel, 2 fh. 3.37^. A Mina, 61. 16 s. 10.5 d. A Talent of filver, 34.2 /. 3 s. 9 d. A Shekel of gold, 1 /. lb s. d. A Talent of gold, 5475 /. 1 A Table [ 278 ] A 'Table Jhewing the Inter eft of any fum of money, from a Million to a Pounds for any number of days, at any rata of Interejl. Sum. 1. *7 9 2.95 ^OOOOO 82 1 1 8 1 0 4 1 .09, 20OOOO 54-7 T 9 1 0 1 0 3-4° ? n t z 73 '9 5 1.70 90000 1 T i I 0.32 80000 219 3 6 0.96 70000 19 1 7 1.59 60000 164 7 8 0.22 500CO 136 l 9 8 2.85 40000 109 11 9 1.48 30000 82 3 10 0.1 1 20000 54 »5 10 2.74 IOOOO 27 7 1 1 l -37 9000 2 4 13 1 3-23 8000 21 18 4 1. 10 7000 *9 3 6 2.96 6000 18 8 9 0.82 5000 13 13 1 1 2.58 4.COO 10 19 2 0.55 3000 8 4 4 2.41 2000 5 9 7 0.27 1 000 2 H 9 2.14 Sum. J. s. d. q. 1000 2 9 2. 14 900 2 9 3 2.12 800 2 3 10 0.1 1 700 1 18 4 1. 10 600 1 12 10 2 80 500 1 7 5 3-7o 400 1 1 11 0,50 300 0 16 5 1 4° 200 0 10 11 2.30 100 0 5 5 3' is 90 0 4 11 5.71 80 0 4 4 2.41 7° 0 3 10 O. i I 60 0 3 3 i-8i 5° 0 2 8 3.51 40 0 2 2 1.21 3° 0 1 7 2.90 20 0 1 1 0.60 10 0 0 6 2.30 9 0 0 5 3-67 8 0 0 5 i-4° 7 0 0 4 2.41 6 0 0 3 3.76 5 0 0 3 4 0 0 2 z.52 3 0 0 1 3.80 2 0 0 1 1.26 1 0 0 0 2.63 Multiply f 279 ] Multiply the fum by the number of days, and the product thereof by the rate of Jntereft per Cent, then cut off the two laft figures to the right hand, and enter the Table with what remains to the left; againft which numbers col- lected, you have the Intereft for the given fum. Example. Qu. What is the Inter ejl of 100/. at 5 per Cent, for 365 days. Number of days 365 Multiply by 100 1. The produfl is 36500 which multiplied by 5 Rate per Cent* makes 1825I00 Then, in the Table, 1. s. d. q. parts. ?icoo is 2149 6.14 800 2 3 10 0.11 20 011 0.00 5 003 ll S l8zs Anfiu. 500 0.00 Juft 5 pounds: and in the fame way the intereft of any other given num- ber of pounds may be found for any given number of days. The decimals are iooth parts of a farthing. A TABLE [ 2 8o ] dumber of <=> *. , 1 Date9 of the feve- al Mint Inden- tures, Standard of the Silver at each Period. shillings, &c. in the found, or iz Ounces Troy, of StandardSil- ver coined at Weight of 20 Shillings in reckoning of Standard Silver, at Weight of fine Silver contain- ed in 20 Shil- lings in rec- koning, at Fine Alloy. each Period. each Period. Silver each Period. A. D. oz. pw. oz. pw. fh. d. oz. pw. gr. oz. pw. gr. CO 1066 I 1 0 18 2 1 A T 1 1 5 0 IO 0 3 I O07 I I 2 0 18 20 O 12 0 0 1 1 2 O I 3CO 1 1 2 0 18 20 3 II 17 1 10 I Q 6 of Eng . D. 17 134-7 I 1 r a -\ 1 .354 J 1395 ^ I I 1 1 2 0 0 18 IS 22 6 0 1 0 9 *3 1 2 g 0 9 8 17 8 17 Id. 1/ "if I4023 I4I 2 1 I 2 0 18 3 2 0 7 10 0 u 18 r 8 v. ^ I 42 2 1 1 2 0 18 30 0 8 0 0 7 8 0 ft vD O 1422 I I z 0 18 37 6 Q O 0 5 18 10 Q t-i 1426 ) l-Q I4465 II 2 0 18 3° 0 8 O 0 7 8 0 «; . 1 46 1 J I404 1 I 482 /» 1 1 2 0 37 c 0 6 8 0 5 18 10 I4»3 1 1 494 J -i r «" <« 1505 1509 / iS3 2 i 1 1 1 1 2 2 0 0 1 0 18 40 45 0 0 6 5 0 6 0 16 5 4 11 0 18 6 .too J 543 IO 0 Z 0 48 0 5 0 0 4- * O 3 8 1545 6 0 6 0 40 c 3 0 0 2 10 0 I54 6 ? 48 7 2 e Jiandar me of W *547 ( 1548J *5#9 4 6 0 0 8 6 0 0 0 0 5 3 0 6 0 16 I I 13 8 13 8 1551 3 0 9 0 72 0 3 6 16 0 16 16 1553 1 1 1 0 19 60 0 4 0 0 3 13 16 1560 | 1583 5 1 1 2 0 18 60 0 4 0 0 3 14 0 ft 1601 " 1605 W 1627 « 1661 ' 1671 ( 11 2 0 18 62 0 3 10 3 11 14" TA 1685 172c J i?6 4 J < C 281 ] jrding to the then Dates of the feve nl Mint Inden- tures. Valueofthe fame 20 Shillings in reckoning of our pre- fent Money. Propor- tion of Money at each Pe- riod to that of ojjr pre- fent /VI 0- ney. Value of the Ounce of the then ftandard Silver to that of our prefent Money. Value of the Ounce of fine Silver at each Period. Kings and Queens in thelc Pe- nods. u 0 A. D. 1. s. d. s. d. s. d. (4 *o IO0O 2 l8 ii * % r 2.g002 c J 2 1 » * 8 wui. v_on<]. O s 1U07 3 2 0 3*1 000 5 2 I Q 5 9i o 3 ^ Will Rtifiic VV 111. KUIUS r? j 1 r lidward 1. Edward III. 0 ~* . 1300 3 I 2I 4- 3 -06 1 4 c J 2 I CJ tn 2 2 i 2 J 1422 2 I 4 „ -AAA 2.C000 2 2 8^ Henry VI. 1 42 2 I 12 w 4 1.0531 c J 2 2 -1- ■ differ ( ; two f 1420 / 14465 2 I 4 2.0666 5 2 2 8£ 1 4 u * lid ward IV. 1464 1.6531 ♦-1 t) 1402 , oi U 4 5 2 j 44 ■ A ft* •4 8 3 hdward V. M *494, tienry VII. P 1505 I 1 1 O i»55°o 2 3 7l 1509 1 153 2 j » 7 61 I -377 6 5 2 4 ©j rlenry vxii* © & 5 543 1. 1635 4. / s 4- 9k CO > 1 0 0 1. 00 00 5 2 5 7 2 1685 James II. •<>» 1720 George I. S *7 6 4- George III. O o From [ aSa ] From A. D. iooo to 1066, price of a horfe \ L ij s. 6 d. — of an ox y s. 6d. — of a cow 6 s. — a fwine 2j. — a fheep 1 j. 3 d. — wheat, per quarter, 1 s. 6d. From A. D. 1066 to 11 99, price of a horfe 12 s. 5 — an ox 4*. 8£d?. — a fow 35. — a colt 2 s. tf-d. — a calf 2 j. 4| df. — a fheep 1 j. 8 af. — wheat, per quar- ter, 3 j. id. From D. 1199 to 1307, a horfe 1 /. 1 1 — an ox 1 /. o j. yd. — a cow 1 7 o^d. — a lamb 4*. — a heifer 2 j. 1^.— a fheep is. "]\d. — agoofe is. old, — acock tf-d.—a. hen $d. — wheat, per quarter, 1/. 3 s. 2\d. From A. D. 1307 to 141 8, an ox 2/. 6s. id.— a. horfe iSs. \d.-~ a cow ys. id. — a calf \s. 2 d. — a fheep 2s. yd.— a goofe 9^. — a cock 3!^. — a hen 2|<^. — ale, per gallon, y\d, — wages of a common day-labourer 4^. — wheat, per quarter, 15*. From A. D. 141 8 to 1524, a horfe 2.1. 4 j.— an ox i f. 15 s. 8~d. — a cow 15 j. 6d. — a colt 7 8^. — a fheep 5 *. 4 * hog [ 28 3 ] a hog 5 x,— a calf 4 i ^ — a cock 3^. a hen 2 ^.—day-labourer's wages 3|^. —ale, per gallon, 2| to ? Value, 2 Value. I 0.9662 — r 26 16.8904 51 1 - 23.6286 76 26.4709 2 1.8997 2" 17-3854 S 2 23.7958 77 26.5506 3 2.8016 28 17.6670 53 2 3.9573 78 26.6190 4 3- 6 73* 29 18.0358 54 24.1133 79 26.6850 26.7488 ' 5 4-jH> 30 l8.3g20 55 24.2641 80 0 5.3286 3l 18.7363 5 6 24.4097 0 I 26.8 1 04 7 6.1145 32 1 9.0689 57 24.5504 82 26.8700 8 6.8740 33 19.3902 58 24.6864 83 26.9275 9 7.6077 34 19.7007 59 24.8178 84 27.983 1 10 8.3166 35 20.0007 60 24.9447 8q 27.0368 11 9.0015 36 20.2905 61 25.0674 86 27.0887 12 9.6633 51 20.5705 62 25.1859 87 27.1388 l 3 10.3027 38 20.841 1 63 25.3004 88 27.1873 H 10.920; 39 21.1025 64 25.41 10 89 27.2341 15 1 1. 5174 4 c 21 3551 6| 25.5178 90 27.2793 16 12.0941 4» 21.5991 66 25.621 1 9 1 27.32,30 17 1 2.65 1 2 42 21.8349 67 25.7209 92 27.3652 18 13.1897 43 22.0627 68 25.8173 93 27.4060 •9 13.7098 44 22. 2828 69 2 : .9 1 04 9 + 27-4454 20 14.2124 45 22.4955 7° 26.0004 95 27.4835 21 [4.6980 46 22.7009 7 1 26.0873 96 27.5203 22 1 5. 167 1 47 22.8994 7 2 26.1713 97 27.5558 23 1 5.6204 48 Z3. O9I 2 73 26.2525 98 27.5902 24 16.0584 49 23.2766 74 26.3309 99 27.6234 2 5 16.4815 5° 23.4556 75 26.4067 100 27.65^4 fie Value of the Perpetuity is z2j Tears Pur chafe. Table [ 288 ] Table III. The prefent V due of an Annuity of one Pounds for any number of years under I oo, Inter eft at 4 per Cent. Value. S> 0.9615 1. 8860 27750 3.6298 4.4518 5.2421 6.0020 6 - 73 2 7 7- 4353 8.1108 8.7604 9.2S50 9.98s 6 10.5631 11.1183 1 1.6522 12.1656 12.6592 I 3-i339 13.5903 14.0291 14.45 n 148568 15.2469 15.6220 Value. 5.9827 6.3295 6.6630 6.9837 7.2920 758 4 7.8775 8.1476 8.41 1 1 8.664.6 8.9082 9 1425 9.3678 9.5844 9.7927 9.9930 20.1856 20.3707 20.5488 20.7200 20.8846 21.0429 21.1951 21.3414 21.4821 Value. 21.6714 2I -7475 53 21.8726 21.9929 22.1086 22^2198 22.3267 22.4295 22,5284 22 6234 22.7148 22.8027 22.8872 22.9685 23.0466 23.1218 23. 1940 23.2635 23.3302 2 3-3945 23.4562 23.5156 2 3-5727 23.6276 23.6804 *< a W • Value. 76 Of 3 ' 1 77 78 23.8268 no 80 2 2 .m e 1 < 37'5.f 8l 22.QC7 I fl ? 0 -• 83 2 3-997 2 24.0357 84 24.0728 85 24.1085 86 24.1428 87 24.1757 88 89 24.2074 2 4 2379 90 24.2672 91 24.2954 92 24.3225 93 24.3486 94 2 4-3736 9 l 24.3977 96 24.4209. 97 24.443 r 98 24.4646 99 24.4851 1 00 24.5049 Tafle [ ««9 ] Table I. 2 be prefent V alue of an Annuity of one Pounds to continue fo long as a Life of a given Age is in beings Inter efl being eftimated at 3 per Cent. > — 1 Value. > Value* D> n — Value. Hi' f."*' > if* Value. I — — 26 . -- 17.50 51 12.26 — 76 4.05 2 16.62 2 7 17-33 5 2 12.00 77 3« 6 3 3 17.83 28 1716 53 ii-7$ 78 3.21 4 18.46 29 1698 54 1 1.46 79 2.78 5 18.90 3° 16.80 55 11.18 80 2-34 6 J9-33 3* 16.62 5 6 10.90 81 1.89 7 19.60 32 16.44 57 ic.61 82 1.43 8 19.74 33 16.25 58 10.32 83 0.96 9 19.87 34 16 06 59 10.03 84 0.49 10 19.87 35 15.86 60 9-73 8s 0 00 11 19.74 36 15.67 6i- 9.42 86 0.00 12 19.60 37 15.46 62 9.11 13 19.47 38 1 5.26 63 8.79 H »9-33 39 15.05 64 8.^0 19.19 40 14.84 65 8.13 16 19.05 41 14.63 66 7-79 17 18.90 42 14.41 67 . 7-4S 18 18.76 43 14.19 63 7.10 19 18.61 44 13.96 69 6.75 20 18.46 45 13-73 70 6.38 21 18.30 46 13.49 71 6.01 22 18.15 47 '325 72 5- 6 3 23 17.99 48 13.01 73 5.25 24 17.8.^ 49 12.76 74 4.85 2<; 17.66 5° 12. qi 75 4-45 P p Table [ 2QO ] Table II. The prefent V alue of an Annuity of one Pounds to continue fo long as a Life of a given Age is in beings Inter ejl being ejiimated at 3^ per Cent, > 1 value. > > > OrQ 1 a Value. n Value. a Value. I 14.16 _ 26 16.28 «_ _ 5 1 11.69 76 3-9 8 2 1 5*53 27 16.13 C 2 J 1 1 .4c 77 3.57 3 16.56 28 15.98 53 1 1 .zo 78 3.16 4 17.09 2 9 I 5« 8 3 54 10.95 79 2.74 5 17.46 10 j 15.68 cc J 1 10.60 80 2.3 1 6 17.82 31 15-53 56 10.44 81 I,S| 7 18.05 3 2 15-37 57 10.18 8z 1.42 8 18.16 33 15.21 58 9-9i 83 0.95 9 18.27 34 15.05 59 9.64 84 0 48 10 18.27 35 14.89 60 9-36 8q o.co 11 1 3. 16 36 14.71 61 9.08 86 0.00 12 18.05 37 14.52 62 8-79 13 17.94 38 >H-34 63 8.49 H 17.82 39 14.16 64 8.19 15 17.71 40 13.98 65 7.88 16 17-59 4 1 13-79 66 7.56 17 17.46 42 13*59 67 7.Z4 18 17-33 43 13.40 68 6.91 '9 17.21 44 13.20 69 6.57 20 17.09 4 5 12.99 70 6.22 21 16.90 46 i 2.78 7 1 5.87 22 16-83 47 12. 7 7 2 5-5i 23 16.69 48 12.36 73 5.14 24 16.56 49 12.14 74 4-77 2, 16.42 so 1 1 .92 75 4.38 Table [ 2 9i ] Table III. The frefent V due of an Annuity of one Pounds to continue fo long as a Life of a given Age is in beings Inter efi being eftimated at 4 per Cent. > Value, > > HQ n> Value. * Value. n> Value. I 13.30 20 15.19 1 5»oo 5 1 11.13 7° 3-9 1 2 14.54 2 7 5 2 10.92 77 3-5 2 3 1 5*43 14.94 53 10.70 7° 3 11 4 15. 39 7 n zy 14.81 54 10.47 7 9 2 .70 r D 16.21 10 14.68 r p 1 0.24 80 2.28 6 16.50 31 14.54 10.71 81 7 16.64 3 2 14.41 57 9 57 82 1.40 . 8- 16.79 33 14.27 58 9.22 83 0.9; 9 16.88 34 14.12 59 9.07 84 0.48 10 16.88 35 13.98 60 971 85 0.00 1 1 16.79 36 13 82 61 8.45 86 0.00 12 16.64 37 13.07 62 8.28 '3 1 6.60 38 *3-5 2 63 8.90 «4 16.50 39 13 3 6 6 4 7.92 15 16.41 40 1 3.20. 7.63 16 I6.31 41 13.02 66 7-33 17 16.21 42 12.85 67 7.02 18 16. 10 43 12.68 68 69 6.71 19 15.99 44 ' 12.50 6.39 20 .5.89 45 12.32 70 6.06 21 I5.78 46 12.13 7i 5.72 22 15.67 47 11.94 72 5.38 ! 2 3 \}$:$s 48 11.74 73 5.02 24 »5-43 49 11.54 74 4.66 11 15.31 50 11.34 75 4.29 p 2 Befides [ 292 ] Befides the ufe of the firft three of thefe Tables, as expreffed by their titles, they ferve likewife to refolve the quef- tions concerning compound Inter eft ; as 1. To find the prefent Value of 1000/. payable 7 years hence, at 3! per Cent. From the prefent value of an Annuity of 1/. certain for 7 years, which, in Table II. is 6. 1 14.5, I fubtradt the like value for 6 years, which is 5.3286; and the remainder .7859 is the value of the 7 th year's rent, or of 1 /. payable after 7 years ; which multiplied by 1000 gives the anfwer 785 /. 18 s. 2. If it be afked, what will be the Amount of the fum S in 7 years at 3! per Cent. ? Having found . 7 8 5 9 as above, 'tis plain the amount will be ~< .3. If the queftion is, In what time a fum S will be doubled, tripled, orincreafed^ in . any given Ratio at 3, 3^ Sec. per Cent. I take in the proper Table two contiguous numbers, whofe difference is neareft neareft the reciprocal of the Ratio given, as |, &c. And the year againft the higher number is the Anfwer. Thus, in Table I. againft the years 22, 23, ftand the numbers 15.9639 and 16.4436 ; whofe difference .5067 being a little more than .5, or \, fhews that in 2 3 Years, a fum S will be a little lefs than doubled, at 3 per Cent, com- pound Intereft. And againft the years 36 and 37 are 21.8323, and 22.1672 ; the difference whereof being ,3349, nearly |, fhews that in 14 years more it will be almoft tripled. If more exa&nefs is required, take the adjoining difference, whofe error is contrary to that of the difference found ; and thence compute the pro- portional-part to be added or fubtra&ed. Thus in the laft of thefe Examples, the difference between the years 37 and 38 is .3252, which wants .0081 of .3333 (== I), as the other difference • 3 349 exceeded it by .00 1 6. The 3 8th year is therefore to be divided in the Ratio of 16 to 81 ; that is |f of a 9 year, [ 2 94 ] year, or about 2 months to be added to the 3 7 years. j. To find at what Rate of Inter eft I ought to lay out a fum S, Jo as it may increafe \ for Inftance, or become ~ S in 7 years* Here the fradion I am to look for among the differences is J, or the de- cimal .75, which is not to be found in Tab. I. or II. till after the limited time of 7 years. But, in Tab. III. the num- bers againft 6 and 7 years give the dif- ference .7599 y and the Rate is 4 per Cent, nearly. So far Mr. De Moivre on this Subject. In queftions concerning the Values of Lives any how combined, recourfe muft be had to Mr. De Moivre $ laft Edition of his Treatife on Annuities. The four following Tables, and the .Remarks on them, are alfo copied from Mr. De Moivre V Book on the Doblrine of Chances. The [ 2 95 ] The Probabilities of JIuman Life, ac- cording to different Authors* Table I. By Dr. Halley. A crp J n & c - ) Living. Age. Living. Living. Ape. Living. I 1000 580 A C oy/ 2 8^5 24 574 11 387 68 162 3 ,798 2 5 567* 47 377 69 152 4 760 26 560 48 367 70 i£2 5 73 2 2 7 553 « 546 49 357 7i 131* 6 710 28 5° 34 6# 72 120 7 692 29 539 51 335 73 109 8 680 3o 53'* 5 2 3 2 4 74 98 9 670 3' 523 53 3i3 n l 88* IO 661 32 5'5 54 302 76 78 1 1 ^53 33 507 55 292* 77 68 12 646 34 499 56 282 78 58 13 640* 35 490* 57 272 79 49* 634 36 481 58 262 80 4i '5 628 37 472 59 292 81 34 16 622 38 4 6 3 60 242 82 28 17 616 39 454 6x 232 83 *3 18 610 40 445 62 222 84 19 19 604 41 436 % 212 # # 20 508 42 427 64 202 21 <92 43 417* 6,5 192 22 ! <;S6 44 407 1 t ' 1 182 Table [ 296 ] Table II. By Mr. Kerjfeboom. Age. o Lfving, 1400 Age. Living. Age. Living. Age. Living. I 1 1 25 26 760 5 l 495 7 6 160 2 1075 27 747 52 482 77 H5 3 1030 20 735 53 470 78 130 4 993 29 7 Z 3 54 458 79 u? 5 964 30 711 55 446 80 100 a o 947 3 l 699 . a 5® 434 81 87 7 93° 3 2 Ao_ 007 57 42 1 82 7J o 5 913 33 6 75 S 8 408 83 64 9 904 34 ' AA ~ OO5 59 395 84 55 IO 895 35 & 55 ' Art OO s 5 ii 886 36 6 45 6l 369 86 11 12 878 37 635 62 35 6 87 28 J3 870 ^8 63 343 88 21 H 863 856 39 64 3 2 9 89 *S 15 40 6> 315 90 10 16 849 41 596 66 3°» 91 7 17 842 42 587 67 1 287 92 5 - 18 835 43 578 68 273 93 3 19 826 44 5 6 9 69 259 94 2 20 817 45 560 70 245 95 1 21 808 46 550 71 z 3» 96 0.6 22 800 47 540 72 217 97 0.5 23 792 48 530 73 203 98 0.4 24 783 49 518 74 189 99 0.2 «5 722 50 5°7 75 *75 100 0.0 Table [ *97 ] Table III. By M. de Parcieux. Ape, Li viihi. ^•2e. Living. 6 ' Living. Liivincr. I *«** 26 766 C T !) * C7 1 3 / 76 IQ2 2 27 7<8 / 3 C 2 c6o 77 Id 2 0 I 000 28 7 CO C X d i 78 I C4 A. i 970 20 y 742 <>4 518 3 3 79 I 26 r 3 04.8 7 2 J. C C !> 5 c 26 80 I 18 6 0 5 0 X I 726 c6 3 U C I 1 3 T 81 82 r 01 7 y 3 52 J) 718 C7 C02 j 8 c 8 902 33 710 58 489 83 71 q y 8go 702 CO 476 84 CO IO 880 35 694 60 4' ) 3 85 48 1 1 872 36 686 6l 450 86 38 I 2 866 37 678 62 437 87 29 13 860 38 671 63 423 88 22 H 854 39 664 64 409 89 l6 15 848 842 40 65 7 65 395 90 I I 16 41 650 65 380 9 1 7 «7 *35 42 643 67 3 6 4 92 4 18 828 43 636 68 347 93 2 19 821 44 629 69 3 2 9 94 1 20 814. 45 622 70 310 95 0 21 806 46 J15 7 1 291 96 # 22 798 47 607 72 271 97 « 23 790 48 599 73 25 1 98 24 782 49 590 74 231 99 2 c. 774 50 581 75 z 1 1 100 Q^q Table [ 2 9 8 ] Table IV. By Meffrs. Smart and Simpfon. \ Age. Living. Age. Liv- Age. Liv- ing Age. Liv- ing nig. Age. Liv- int> . i 1 280 I 870 !" 17 480 33 358 49 212 '65 99 2 700 18 474 34 349 5° 204 66 93 3 6 35 19 468 35 34° 196 l 6 7 87 4 600 20 462 36 33' 5' 2 188 68 81 5 580 21 455 37 322 53 180 :^9 75 6 564 22 44 8 38 313 54 172 70 69 7 55i 2 3 44 1 39 3°4 55 165 71 64 8 54' 24 434 40 294 S6 158 1 72 59 9 532 25 42b 4' 284 57 15' 73 54 IO 5 2 4 26 418 42 274 58 144 ■ 74 49 1 1 517 27 410 43 264 59 137 75 45 12 510 28 402 44 2 55 60 130 , 7<> 4i 13 504 29 394 45 246 61 123 77 78 38 H 498 30 385 46 2 37 62 117 35 I 5 49 2 3 1 376 47 228 63 11 1 79 3 2 16 486 32 3 6 7 48 220 64 ic 5 80 29 Remarks on thefe four 'Tables of the Pro- babilities of Human Life, The firft Table is that of Dr. Halley* compofed from the Bills of Mortality of the city of Breflaw ; the beft, perhaps, as well as the firft of its kind : and which will always do honour to the judgment [ 299 ] judgment and fagaclty of its excellent author. The next is a Table of the ingenious Mr. Kerjjeboom, founded chiefly upon Regifters of the Dutch Annuitants, carefully examined and compared for more than a century backward. And Monfuur de Parcieux, by a like ufe of the lifts of the French Tontines, or long Annuities, has furnifhed us Table III; whofe numbers were likewife verified upon the Necrologies or mortuary Re- gifters of feveral religious houfes of both fexes. To thefe is added a Table of Meffieurs Smart and Simp/on adapted particularly to the city of London ; whofe inhabi- tants, for reafons too well known, are fhorter lived than the reft of mankind. Each of thefe Tables may have its particular ufe : the fecond or third in valuing the better fort of lives, upon which one would choofe to hold an Annuity ; the French may ferve for London, or for lives fuch as thofe of its inhabitants may be fnppofed to be : Qji 2 while [ 3oo ] while Dr. Halley\ numbers, falling be- tween the two extremes, feem to ap- proach nearer to the general courfe of nature. And in cafes of combined lives, two or more of the Tables may perhaps be ufefully employed. Befides thefe, the celebrated Monfieur de Buff on * has lately given us a new Table, from the adual obfervations of Monfieur du Pre de St Maur of the French Academy. This Gentleman, in order to flrike a juft mean, takes three populous paddies in the city of Paris, and fo many country villages as furnifh him nearly an equal number of lives : and his care and accuracy in that per- formance has been fuch as to merit the high approbation of the learned editor. It was therefore propofed to add this Table to the reft ; after having cleared its numbers of the inequalities that ne- ceflarily happen in fortuitous things, as well as thofe arifing from the carelefs manner in which Ages are given to the Parifh Clerks ; by which the years that i Hifieire Naturelle^ Tom. II. are [ 3 GI 3 are multiples of 10 are generally over- loaded. But this having been done with all due care, and the whole reduced to Dr. Halleys denomination of i ooo infants of a year old, there refulted only a mu- tual confirmation of the two Tables ; Mr. du Pres Table making the lives fo me what better as far as 39 years, and thence a fmall matter worfe than they are by Dr, H alley s. We may therefore retain this laft as no bad ftandard for mankind in general \ till a better police, in this and other nations, fhall furnifli the proper Data for correcting it, and for expreffing the Decrements of life more accurately, and in larger numbers. A Table; [ 3<>2 ] A Table ; the fir ft part whereof Jhews the height to which a Barometer muft be raifed above the plane fur face of the Earth, in order that the Mercury may ft and at any given height in the Tube ; and the fecond part Jhews at what height the Mercury will ft and in the Tube, when the Barometer is raifed to any given height above the Earth's plane furface. Part i. Height of the Mercury in inches 30.000 29.000 28.000 27.000 26.000 25.000 20.000 1 5. 000 10.000 5.000 1.000 0.5 0.25 0.1 0.001 0.000 Height of the Barome- ter in feet above the Earth's plane furface. 29 miles, 41 miles, 53 miles Feet o 9'5 1862 2844 3863 4922 10947 18715 29662 4S378 9183 1 n 05 47 1 29262 1 53 120 2 16480 279840 Part JI. Height of the Barom- eter above the Earth. Height of the Mer- cury in inches. Feet 0 30,00 1000 28.91 2000' 27.86 3990 26.85 4000 25.87 5000 H-93 Miles 1 24.67 2 20.29 3 16.68 4 I3-72 5 11.28 10 4.24 20 1 .60 2 5 0.95 3° 0.23 40 •0.08 By [ 303 ] By the firft part of this Table, and a common Barometer or Weather-glafs, the perpendicular height of a hill above the plane furface of the Earth, may be nearly found. Thus, fuppofe the Mer- cury wasobferved to ftand at 30 inches in the tube when at the foot of the hill, and at 27 inches when carried up to the top : againft this finking of three inches, you have 2844 feet (or 948 yards) for the perpendicular height of the hill. The fecond part is too plain to need any defcription or example. Anaccount of M. Villette "s concave burn- ing Mirror. This Mirror is 3 feet 1 1 inches in diameter, and its focal diftance is 3 feet 2 inches. It is made of copper, tin, and bifmuth. The effect of the Sun -beams on dif- ferent bodies held in its focus were as follows : A piece 4 [ 3°4 ] A piece of Roman tile began to melt in 3 feconds* and was ready to drop in 1 00 feconds. Chalk fled away in 33 feconds. A foflil (hell calcined in 7 feconds. Copper ore vitrified in 8 feconds. Iron ore melted in z \ feconds. A g-eat tooth of a fifh melted in 33 feconds. "Welfh afbellos was a little calcined in 28 feconds. A king George's halfpenny melted in 16 feconds. Tin melted in 3 feconds, and had a hole in it in 6. A bone calcined in 4 feconds, and was vitrified in 33. A diamond weighing 4 grains loil | pares of its weight. The folar beams are condenfed 1700 times in the focus of this mirror (the condenfation in the focus being as the area of the mirror is to the area of its focus) and their heat, in the focus, is, 433 times as great as the heat of com- mon fire. The proportional breadth of each colour in the Rain-bow ', fuppofing the whole breadth thereof to be divided into 360 equal parts. The red, 45 parts; the orange, 27; the yellow, 48 ; the green, 60 ; the blue, 60 ; the indigo, 40 ; and the violet, 80. If the fiat upper furface of a top be divided into 3 60 equal parts, all around its [ 3°S ] its edge, and be divided by 7 lines into fo many portions or fe&ors of circles, in the above proportions, and the re- fpec~Uve colours be lively painted in thefe fpaces, but fo as the edge of each colour may be made nearly like the colour next adjoining, that the repara- tion may not be well diftinguifhed by the eye ; and the top be made to Ipin, all thefe colours together will appear white. And if a large round black fpot be painted in the middle, fo as there may be only a broad flat ring of colours around it; the experiment will fucceed the better. Red is the leafi. refrangible of all co- lours, orange the next leaf!:, yellow the next, green the next, blue the next, in- digo the next, and violet the mod of all. Mr. Edward Delaval, F; R. S. has found, by experiments on melting dif- ferent metals with pure glafs, that they colour the glafs according to their differ- ent den fi ties or fpecific gravities \ the inoft denfe giving a red colour to the R r glais a [ 3°6 ] glafs, and the leaft a blue or violet, Thus, gold melted with glafs makes it red ; lead melted with glafs, gives it an orange colour; diver a yellow j copper a green ; and iron a blue. Colours produced by the mixture of co~ lourlefs fluids* Spirit of wine mixed witli fpirit of vitriol make a red. Solution of mercury mixed with oil of tartar, orange. Solution of fublimate and lime-water, yellow. Tincture of rofes and oil of tartar, green. Solution of copper and fpirit of fal-armoniac, purple. Tinfture of rofes and fpirit of wine, blue. Solution of fublimate and fpirit of fal-armoniac, 7 ] Tincture of red rofes, which is red, and fpirit of hartfhorn which is brown- ifh, make a blue. Tincture of violets, which is blue, and folution of Hungarian vitriol, which is blue, make a purple. Tincture of violets, which is blue, and folution of copper, which is green, make a violet* Tincture of cyanus (blue-bottle flow- er) which is blue, and fpirit of fai ar- moniac coloured blue, make a green. Solution of Hungarian vitriol, which is blue, and lixivium, which is brown, make a yellow. Solution of Hungarian vitriol, which is blue, and tincture of red rofes, make a black. Tincture of cyanus, which is blue, and volution of copper, which is green, make a red. Colours changed^ and rejlored. Solution of copper, which is green^ by fpirit of nitre is made colourlefsj and is again reftored by oil of tartar. R r 2 Limpid C 308 ] Limpid infufion of galls is made black by a folution of vitriol, and tranfparent again by oil of vitriol ; and then black again by oil of tartar. Tin&ure of red rofes is made black by a folution of vitriol, and becomes red again by oil of tartar. A flight tincture of red rofes, by fpirit of vitriol becomes a fine red ; then, by fpirit of fal armoniac turns green ; and then, by oil of vitriol becomes red again. Solution of verdigreafe, which is green by fpirit of vitriol becomes co- lour! efs ; then, by fpirit of fal armoniac becomes purple ; and then, by oil of vitriol becomes colourlefs again. CD 7*he quantity of Land and of Water on the Earttis fur face. The feas and unknown parts of the Earth (by a meafurement of the beft maps) contain 160,522,026 fquare miles; the inhabited parts 38,990,569: Europe, 4,45 6,0 65 j Aria, 10,768,823; 7 Africa, [ 3°9 ] Africa, 9,564,807; and America, 14,110,874. In all, i99>5 12 >595 ; which is the number of fquare miles on the whole furface of the Earth. The weight of the whole Atmofphere. On a fquare inch, it is 15 pounds ; on a fquare foot, 2160; on a fquare yard, 19,440; on a fquare mile, 60,217,344,000; and on the whole furface of the Earth, and Sea together, 1 2,014,118,565,447,680,000 pounds. The furface of the body of a middle fized man is about 14 fquare feet ; and as the weight or preffure of the air is equal to 2160 pounds on every fquare foot on (or near) the Earth's furface ; and as the preffure of the air is equal in all manner of directions, its preffure on the whole body of a middle fized man is equal to 30,240 pounds, or 13^ tons. But, becaufe the fpring of the internal air is of equal force with the preffure of the external, the preffure is not felt. [ 3io ] The diameter and circumference of the vifible part of a cloudy Jky. The greateft diftance of the clouds in the horizon at fea is 94 miles from the obferver, all around; and eonfe- quently, the whole extent or diameter of the horizon, reaching to the clouds, is 188 miles; and the circumference thereof is 590.97 miles- The velocity of Light. It has been proved* by the eclipfes of Jupiter's Satellites* that light takes 8 minutes of time to come from the Sun to the Earth. And as the Earth's diftance from the Sun is 95,000,000 miles in round numbers, 'tis plain that the velocity of light is 11,875,000 miles in a minute* and confequently 197,916 miles in a fecond; which is 1,4.86,458 times as fwift as the motion of a cannon ball, and 10,440 times as fwift as the Earth moves in its annual orbit. The [ pi ] The velocity of found. According to Dr. Halley, Mr. Flam- fleed> and Mr. Derham, found moves 1142 feet in a fecond of time, 68,520 feet in a minute, and 4,1 1 1,200 miles in an hour. Hence we may know how far a thun- der cloud is from us, if we have a watch that fhews feconds. Thus, fuppofe there were four feconds from the mo- ment we fee the flam of lightning to the moment we hear the clap of thwider, 'tis plain that the cloud which produced the thunder is four times 1142 feet, or 4568 feet from us; which is about four-fifths of a mile. "The caufe of the ebbing and flowing of the fea at the fame time on oppofte fdes of the globe. The reafon why the tides rife on the fide of the Earth which is at any time turned towards the Moon, is plain to every one \ becaufe her attraction muft occalion [ S 1 * ] occafion a fwelling of the waters to- ward her on that fide : but the caufe of as great a fwell, at the fame time, on the oppolite fide of the Earth, which is then turned away from the Moon, has been very hard to account for ; he- caufe the rifing of the tide there is in a direction quite contrary to the attraction of the Moon. . , But this difficulty is immediately removed, when we con- fide^ that all bodies moving in circles have a centrifugal force, or conftant tendency to fly off from the centers of the circles they defcribe ; and this cen- trifugal force is always, in proportion to the diftance of the body from the center of its orbit, and the velocity with which it moves therein. When the body is large, the fide of it which is fartheft from the center of its orbit will have a greater degree of centrifugal force than the center of the body has ; and the fide of it which is neareft the center of its orbit will have a lefs degree of centrifugal force than its center has. As [ 3i3 ] As the Moon goes round the Farth every month in her orbit, the Earth alfo goes round an orbit every month, which is as much lefs than the Moon's orbit, as the quantity of matter in the Moon is lels than the quantity of matter in the Earth, which is 40 times. For, by the laws of nature, when a fmall body moves round a great one, in free and open fpace, both thefe bodies muft move round the common center of gra- vity between them. The Moon's mean diftance from the Earth's center is 240,000 Engl ifli miles: divide therefore this diftance by 40, the difference between the quantity of matter in the Earth and Moon ; and the quo- tient will be 6000 miles, which is the diftance of the common center of gra- vity (between the Earth and Moon) from the center of the Earth. Now, as the Earth and Moon move round the common center of gravity between them, once every month ; 'tis plain, that whilft the Moon moves round her orbit, at 240,000 miles from the S f Earth's [ 314 ] Earth's center, the center of the Earth defcribes a circle of 6000 miles radius, round the center of gravity between the Earth and the Moon ; the Moon's at- traction balancing the centrifugal force of the Earth at its center. The diameter of the Earth is 8000 miles, in round numbers, and confe- quently its femidiameter is 4000 : fo that the fide of the Earth, which is at any time turned toward the Moon, is 4000 miles nearer the common center of gravity between the Earth and Moon than the Earth's center is ; and the fide of the Earth, which is then fartheft from the Moon, is 4000 miles farther from the center of gravity between the Earth and Moon than the Earth's center is at that time. Therefore, the radius of the circle defcribed by the parts of the Earth which come about toward the Moon, by the Earth's diurnal motion, is scoo miles; the radius of the circle defcribed by the Earth's center is 6000 ? and the radius of the circle defcribed by thofe parts [ 3*5 ] parts of the Earth which, irr revolving on its axis, are furtheft from the Moon, is 10,000 miles. The centrifugal forces of the differ- ent parts of the Earth being directly as their diftances from the abovementioned common center of gravity, round which both the Earth and Moon move, thefe forces may be expreffed by 2000 for the fide of the Earth nearefl: the Moon, by 6000 for the Earth's center, and by 10,000 for the fide of the Earth which is fartheft from the Moon. But the Moons attradion is greateft on the fide of the Earth next her, where the centrifugal force or tendency to fly off* from the common center of gravity (and confequently, from the Moon) is leaft ; and therefore, the tides muft rife on the fide of the Earth which is nearefl: the Moon, by the excefs of the Moon's attraction. As her attraction balances the cen- trifugal force at the Earth's center, 'tis plain^that the centrifugal force of the fide of the Earth which is fartheft from S f 2 [ 3i6 ] the Moon is greater than her attraction ; and therefore, the tides will rife as high upon that fide from the Moon, by the excefs of the centrifugal force, as they rife on the fide next her by the excefs of her attraction. And as the Earth is in conftant motion on its axis, fo as that any given meridian revolves from the Moon to the Moon again in 24 hours, minutes, each place will come to the two eminences of water, under and oppofite to the Moon, in 24 hours, 50' minutes, or have two tides of flood and two of ebb in that time. For, as much as the waters rife above the common level of the furface of the fea, under and oppofite to the Moon,, fo much they rauft fall below that level half way between the higheft places; or at 9a degrees from them. On thefe principles, it is equally eafy to account for the rifing of the tides, at the fame time, on both fides of the Earth : and this rifing is made evident to fight in my Lecture on the central forces j and the principles on which it depends [ 3'7 ] depends are made obvious to the under- ftandings of all the obfervers. Surprifing properties of numbers^ placed in fquares and circles. I have feen feveral different kinds of (what is generally called) magic fquares ; but have lately got a magic fquare of fquares and a magic circle of circles of a very extraordinary kind, from Dr. Benjamin Franklin of Philadelphia^ with his leave to publifli them. The magic fquare goes far beyond any thing of the kind I ever faw before ; and the magic circle (which is the firft of the kind I ever heard of, or perhaps any one befides) is ftill more furprifing. What the Doctor's rules are, for difpofing of the different numbers fo, as that they fhall have the following properties, I know nothing of : and perhaps the rea- fon may be, that I have not ventured to afk him ; although I never faw a more communicative man in my life. The plates of thefe are at the end of the book. . Plate C 318 ] Plate I. A magic fquare of fquarzs* The great fquare is divided into 256 fmall fquares, in which all the numbers from 1 to 256 are placed, in 16 columns, which may be taken either horizontally or vertically. The proper- ties are as follows % 1. The fum of the fixteen numbers in each column, vertical and horizontal, is 2056. 2. Every half column, vertical and horizontal, makes 1028, or half of 2056. 3. Half a diagonal afcending, added to half a diagonal defcending, makes 2056 ; taking thefe half diagonals from the ends of any fide of the fquare to the middle thereof; and fo reckon- ing them either upward, or downward; or fidewife from left to right hand, or from right to left. 4. The fame with all the parallels to the half diagonals, as many as can be a drawn [ 3*9 ] drawn in the great fquare: for any two of them being directed upward and downward, from where they begin to where they end, their fums will make 2056. The fame downward and up- ward from where they begin to where they end ; or all the fame if taken fide- ways to the middle, and back to the fame fide again. JV. B. One fet of thefe half diagonals and their parallels, is drawn in the fquare upward and downward. Ano- ther fuch fet may be drawn from any of the other three fides. 5. The four corner numbers in the great fquare added to the four central numbers therein, make 1028 ; equal to the half fum of any vertical or horizon- tal column, which contains 1 6 numbers ; and equal to half a diagonal or its pa- rallel. 6. If a fquare hole (equal in breadth to four of the little fquares) be cut in a paper, through which any of the fix teen little fquares in the great fquare may be feen, and the paper be laid on the great C 320 ] great fquare ; the fum of all the 16 numbers, feen through the hole, is equal to the fum of the lixteen numbers in any horizontal or vertical column, viz. to 2056, Plate II. A magic circle of circles. This circle is compofed of a feries of numbers, from 12 to 75 inclufive, di- vided into eight concentric circular fpaces, and ranged in eight radii of numbers, with the number 1 2 in the center ; which number, like the center, is common to all thefe circular fpaces, and to all the radii. The numbers are fo placed, that the fum of all thofe in either of the con- centric circular fpaces above-mentioned, together with the central number 1 2, make 360 ; equal to the number of degrees in a circle. The numbers in each radius alfo, to- gether with the central number 12, make juft 360. The [ 3 21 ] The numbers in half of any of the above circular fpaces, taken either above or below the double horizontal line, with half the central number 1 2, make 180 ; equal to the number of degrees in a femicircle. If any four adjoining numbers be taken, as if in a fquare, in the radial divifions of thefe circular fpaces ; the fum of thefe, with half the central num- ber, make 180. There are, moreover, included four fets of other circular fpaces, bounded by circles which are excentric with refpect to the common center ; each of thefe fets containing five fpaces. The centers of the circles which bound them are at A y By Cy and D. The fet whofe center is at A is bounded by dotted lines ; the fet whofe center is at G is bounded by lines of fhort unconnected ftrokes; and the fet round D is bounded by lines of unconnected longer ftrokes, to diftinguifh them from one another. In drawing this figure by hand, the fet of concentric circles fhould be drawn T t with [ 322 ] with black ink ; and the four different fets of cxcentric circles with four kinds of ink of different colours; as blue, red, yellow, and green, for diftinguifhing them readily from one another. Thefe fets of excentric circular fpaces interfeel thofe of the concentric, and each other: and yet, the numbers con- tained in each of the excentric fpaces, taken all around through any of the 20, which are excentric, make the fame fum as thofe in the concentric; namely 360, when the central number 12 is added. Their halves alfo, taken above or below thedouble horizontal line, with half the central number, make 180. Obferve, that there is not one of the numbers but what belongs at leaft to two of the circular fpaces ; fome to three, fome to four, fome to five : and yet they are all fo placed, as never to break the required number 360, in any of the 28 circular fpaces within the primitive circle. To bring thefe matters in view, 1 have taken out ail the numbers as above- [ 323 ] above-mentioned : and have placed them in feparate columns, as they ftand around both the concentric and excen- tric circular fpaces, always beginning with the outermoft, and ending with the innermoft of each fet ; and alfo the numbers as they ftand in the eight radii, from the circumference to the center : the common central number 12 being placed the loweft in each column. i . In the eight concentric circular fpaces. H 7 2 2 3 65 2 ( 67 12 74 2; 63 16 7° 18 68 27 61 3° 56 39 49 37 5i 2o 58 47 3 2 54 34 52 43 45 40 55 33 53 35 44 42 57 3 1 48 38 5° 36 59 29 62 24 71 17 69 19 60 26 73 '5 22 66 20 75 13 1 2 12 12 I 2 12 12 12 12 360 360 360 360 360 360 360 360 2. In the eight radii. 14 2 5 3° 41 46 57 62 73 7 2 63 56 47 40 3 1 24 1 5 23 16 39 3 2 55 48 7i 64 65 70 49 54 33 3§ 17 22 3 l 18 37 34 53 5 £ 69 66 67 68 5 1 5 2 35 3 6 J 9 20 1 2 27 28 43 44 59 60 75 74 61 58 45 42 29 26 13 12 12 12 12 12 1 2 1 2 1 2 360 360 360 360 360 360 360 3 6o [ 324 ] *4 3 2 2 3 65 21 02 0 16 70 18 68 39 49 37 5 1 28 54 34 5 2 43 45 33 53 35 44 42 4 8 38 50 36 59 24 71 1 7 69 I Q 73 15 j 64 22 65 1 2 1 2 1 2 1 2 1 2 360 360 260 360 260 3° 5 6 39 49 37 47 T/ 32 C4. 1 T 2 A o4 r 2 3 55 33 53 35 44 3 8 5° 36 59 29 J 7 69 19 60 26 64 22 » 66 20 7C 72 23 65 2 1 67 25 63 16 70 t8 12 12 12 12 1 2 360 360 360 360 360 46 40 55 33 53 D 48 28 r 0 26 7 1 17 69 '9 60 22 DO 20 75 13 65 21 67 1 2 74 16 70 18 68 27 56 39 49 37 5 1 41 47 3 2 54 34 12 12 .- 12 12 12 360 360 360 360 360 62 24 7i 17 59 I C 3 64. 22 66 2 3 65 21 67 12 7° 18 68 27 61 49 37 5 1 28 58 3 2 -54 34 5 2 43 40 55 33 53 35 57 3' 48 38 5° 12 1 2 12 12 12 360 360 360 360 360 [ 325 3 If now, we take any four 14 72 numbers, as in a fquare form, 2563 either from N° 1. N° 2. (as 6 fuppofe from N° 1.) as in the margin ; and add half the cen- tral number 1 2 to them, the fum will be 180; equal to half the numbers in any circu- lar fpace, taken above or be- low the double horizontal line: and equal to the number of degrees in a femicircle. Thus, 14, 72, 25, 63, and 6, make 180. A Lift [ 326 ] A Lift of the Apparatus on which Mr. Fergufon reads his Courfe of twelve Leclures on Mechanics y Hydroflatics^ Hydraulics Pneumatics^ Dialing and Aflronomy. The numbers relate to the Lectures read on the machinery to which they are prefixed. I. Simple machines for demonflrating the powers of the lever, the wheel and axle, the pullies, the inclined plane, the wedge, and the fcrew. A compound engine in which all thefe fimple machines work together. A working model of the great crane at Bri/lol, which is reckoned to be the beft crane in Europe. A working model of a crane that has four different powers, to be adapted to the different weights intended to be raifed : invented by Mr. Fergufon. i Apy- [ 3*7 1 A pyrometer tfiat makes the expan- fion of metals by heat vifible to the 90 thoufandth part of an inch ; fo as to be feen by the bare eye at two feet di- ftance from the machine. II. Simple machines for {hewing the center of gravity of bodies, and how far a tower may incline without danger of falling. A double cone that feemingly rolls up-hill of itfelf, whilft it is actually de- fcendirig. A machine made in the figure of a human creature, that tumbles backward, by continually overfetting the center of gravity. Models of wheel-carriages ; fome with broad wheels, others with narrow ; fome with large wheels, and others with fmall : for proving experimentally which fort is the beft. A machine for fhewing what degree of power is fufficient to draw a loaded cart [ 328 ] cart or waggon up-hill ; when the quantity of weight to be drawn up, and the angle of the hill's height, are known. A machine for diminifhing friction ; and fhewing that the fri&ion is not in proportion to the quantity of the fur- face that either rubs or rolls ; but in proportion to the weight with which the machine is loaded. A model of a in oft curious filk-ree], invented by Mr. V errier near Wrington in Somerfetjhire. A large working model of a water- mill for fawing timber. A model of a hand-mill for grinding corn. A model of a water-mill, for win- nowing and grinding corn, drawing up the facks, and boulting the flour. A machine for demonstrating that the power of the wind, on wind- mill fails, is as the fquare of the velocity of the wind. A model [ 329 ] A model of the engine by which the piles were driven for a foundation to the piers of Weflminjler bridge. /■; r 'M: ■ IIL 4 A machine for (hewing that fluids weigh as much in their own elements as they do in air. A machine for fhewing that, on equal bottoms, the preflure of fluids is in proportion to their perpendicular heights ; let their quantities be ever fo great or ever fo fmall. Machines for fhewing that fluids prefs equally in all manner of dire&ions. A machine for (hewing- how an ounce of water in a tube may be made to raife and fupport fixteen pounds weight of lead. A machine for fhewing, that, at equal heights, the fmalleft quantity of water whatever will balance the greateft quan- tity whatever, if the columns join at bottom. U u A ma- [ 33° ] A machine for (hewing how folid lead may be made to fwim in water, and the lighted wood to fink in water. Machines for (hewing and demon* ftrating the hydroftatical paradox. A machine for demonftrating that the weight of the quantity of water difplaced by a £hip is equal to the whole weight of the (hip and cargo. Machines for (hewing the working of fyphons, and the 'Tantalus 's cup. A large machine for (hewing the caufe and explaining the phenomena of ebbing and flowing wells, and of inter- mitting and reciprocating fprings. IV. Machines for (hewing that when folid bodies are immerfed andfufpended in fluids, the folid lofes as much of its weight as its bulk of the fluid weighs ; and that the weight loft by the folid is imparted to the fluid. A hydroltatic balance, for (hewing the fpecific gravities of bodies, and de- tecting counterfeit gold or (ilver. 7 A working C 331 ] A working model of Archimedes 's fpiral pump. Glafs models for fhewing the ftruc- ture and operations of fucking, forcing and lifting pumps. A working model of a quadruple pump-mill for railing water by means of water turning a wheel. A working model of the Perfian wheel for railing water. A model of the great hydraulic engine under London bridge, that goes by the tides, and xaifes water by forcing pumps. A working model of Mr. Blakeys intended engine for railing water by means of fire. V. and VI. An air-pump, with a great appara*- tus to it, for experiments fhewing the weight and fpring of the air, • vn. J;!; An electrical machine, with a very large apparatus, for fhewing a great U u 2 variety [ 332 ] variety of curious and entertaining ex- periments, many of which are entirely new. VIII. A whirling table, for explaining and demonftrating the laws by which the planets move, and are retained in their orbits : that the Sun and all the planets move round their common center of gravity : that the Earth and Moon move round their common center of gravity once every month : that the Earth moves round the Sun, in common with the reft of tlje planets, and turns round its own axis : that the power of gravity diminifhes in proportion as the fquare of the diftance from the attracting body increafes : that a double velocity in any orbit would require a quadruple power or gravity to retain the body in that orbit : that the fquares of the periodi- cal times in which the planets move round the Sun are in proportion to the cubes of their diftances from the Sun. A plain experimental demonftration of [ 333 ] the doclrine of the tides ; and the caufe of their riling equally high, at the fame time, on oppofite fides of the Earth, IX, X, XI, and XII. A machine for fhewing the motions of the comets. An Orrery, fhewing the real motions of the planets round the Sun ; the ap- parent ftations, direct and retrograde motions of Mercury and Venus, as feen from the Earth : the different lengths of days and nights, and all the viciffi- tudes of feafons, arifing from the di- urnal and annual motions of the Earth: the motions and various phafes of the Moon : the Harveft-moon : the tides : the caufes, times and returns of all the eclipfes of the Sun and Moon: the eclipfes of Jupiter's fatellites, and the phenomena of Saturn's ring. In London, any number of perfbns, notlefs than twenty- five, who will fub- fcribe one Guinea each, may have a courfe [ 33+ ] courfe of twelve Le&ures read on the above-mentioned Apparatus, provided they agree to have at leaft three Lec- tures a week ; in which they may ap- point the days and hours that are mod convenient for themfelves. Within ten miles of London, any number, not lefs than thirty, may have a courfe ; each fubfcriber paying one Guinea. And, Within an hundred miles of London, any number of fubfcribers, not lefs than fixty, may have a courfe ; each pay- ing as above. FINIS. ft. Franklin uiv. J.Ferait&orL ilelin, . J.JlfifnRe-J'c BOOKS publijhed by the fame Author. I. A Stronomy explained upon Sir Ifaac Newton\ jT\ Principles, and madeeafy to thofe who have not ftudied Mathematics. To which is added, the method of finding the Planets from the Sun, by the Tranfit of Venus over the Sun's Difc in the year 1761. Thefe Diftances deduced from that Tranfit ; and an Account of Mr. Horrox's Obfer- vations of the Tranfit in the Year 1639: Illuf- trated with 18 Copper-plates. A New Edition, 8vo. 9J. II. An Eafy Introduction to Aftronomy, for young Gentlemen and Ladies: Defcribing the Figure, Motions, and Dimenfions of the Earth ; the different Seafons; Gravity and Light; the Solar Syftem ; the Tranfit of Venus, and its Ufe in Aftronomy; the Moon's Motion and Phafes; the Eclipfes of the Sun and Moon ; the Caufe of the Ebbing and Plowing of the Sea, &c. Second Edition, Price $s. III. An Introduction to Electricity, in Six Sec- tions, r. Of Electricity in general. 2. Defcrip- tionofthe Electrical Machine. 3. A Defcription of the Apparatus (belonging to the Machine) for making Electrical Experiments. 4. How to know if the Machine be in good Order for performing the Experiment, and how to put it in Order if it be not. 5. How to make the Electrical Experi- ments, and to preferve Buildings from Damage by Lightening. 6. Medical Electricity. Illuftrated with Copper-plates, 4*. IV. Lectures on Select Subjects in Mechanics, Hydroftatiss, Pneumatics, and Optics; with the Ufe of the Globes, the Art of Dialling, and the Calculation of the mean Times of New and Full Moons and Eclipfes, ys, 6d.