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 tev/ / 4C7 
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 r% -XfW' 4 . CtfZfik <Tj v / v/ 1 > 
 
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 ^ ^ ft. frtfr'' 'Y\ 
 
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 ^r-r • * 'r 
 
 
 
 
 
 

PLAT K 
 
 1L. 
 
FLAT F. Ill . 
 
j Ivc'Ujx^ {rzrprf^ 
 
 4 cru*^ (AyCc^Uj^ IJvoot^jL. csj uc>tx Jf ^ 
 0**M~ h >yt^tAAj ^cUtcp^ . 5V 
 Olu-^a^U ^>cceA tx^nn^. J sfc*u<_ 
 
 jUi/Jr^ c i^VLc^C^/'Cs ft^vx^j fic^> \j~Qsf~J--<— 
 
 Cvi>Cii^^c^' t>m. 
 
 r/y 
 
 n o t bo ( found -- to d a pr o e a ttt) jjiit shall keep in recollec¬ 
 tion the prudent advice of Helicane, and 
 
 With patience bear 
 
 Such griefs as I do lay upon myself. 
 
 hnA( i> fcTCftc y £erttj too~Ch/fo 
 
 jttdje? fnm+ (**■ y 
 
 - tfu+Jr, vie* face " ' 
 
 Ui^Uujkl ffafaT, j^cce^x)^ jU^yVew' ‘zdctzry'y /X- 
 
 tyV^A^sl j Sft-P 
 ’ 0*4*614** '*'* ^ Xrr 2<^/1 txstly £A~- 
 
 fiu /r-cscrfC ^ j **4^ (y • 
 
T 7 u: dVfccj' ty h<Lj 
 
 Spx -ul !iyic*s&. c*o£- It- ccUcv z, 
 
 • 4 
 
 PREFACE. 
 
 velopn 1 ent of the argume 
 as a n jans of mental di 
 nrrogai ce and exaggerati 
 troyed 
 
 its efficiency. C 
 
 it in favour of this science 
 cipline, guarded fnam the 
 >n which liave so/often des- 
 An exposition' of Mathe¬ 
 matical logic, and the true principles of Science, 
 illustrating the value of he one apu the simplicity 
 <-f llu; liilitjr. fz The en bvcemerit/of Natural Geo¬ 
 metry— mating on » th a foundation o f t he oommo n 
 
 unrlnrstmulmg , distinguishing its Beauties and Uses, 
 adapting it to the wants of the many, to the capacities 
 of the young, to the aptitudes of the uninitiated, and 
 the exigencies of men of business—the people of little 
 time and a definite purpose. 
 
 -frm: 
 
 The word euclid, on the title page, contains 
 the Diagrams of six of the most interesting and cele¬ 
 brated Propositions of Euclid. The number of each 
 Proposition, and the number of the Book in which it 
 is found, are placed at the foot of each letter. The 
 Diagrams in e, u, c, occurring in the First Book, 
 4 the reader will discover, in the following pages, to 
 what they relate. The Diagram in i^is employed 
 to prove that all the angles that can be made in a 
 semicircle are right angles. The Diagram in e- 
 longs to the Proposition respecting the proportion 
 between triangles and parallelograms of the same alti¬ 
 tude. The Diagram in D^is the geometrical founda¬ 
 tion, of the Rule of Three, -k- seemed tlig most-ep- 
 p rnpvinto tri but e tu EucfaHu 1 - associate his v o rk s w i th 
 
 inc numb 
 
Jyj, a f" t/ f i j lfuc( i f' ^ 4 <) 
 
 CONTENTS. 
 
 7 ^ 3 / — ''f^O C A^ 
 
 INTRODlJCTION TO TIIE^ 
 
 Chap. L — Tnfliinnw . nf n xrf f xi n p wp nlnw l\ . fmrim 
 QTtlf * h" Pnwwt < 4 ' -4-. f(n - »i))»» ill 4 K . nc t« w l. on .I MfU l icmo tirS 
 in p'lrtinulfVf 1 
 
 Chap. 2 .—IsEpartxnl Distinctions pertaining to Mathematics. 
 
 Chap. 3.—History of tlir liiwiand A*i i og,i | e. ' jt 4 a^ r » t ' iic Mathe¬ 
 
 matics. 
 
 Chap. 4.—Utility of Mathematics as a means of extending 
 our knowledge of the physical world. 
 
 Chap. 5.—Mathematics as a means of Mental Discipline. 
 Chap. 0.—The Logic of Euclid. 
 
 Chap. 7.—Natural Geometry. 
 
 Chap. 0.—Practico and Theory, or the Distinction between 
 Practical Geometry and Pure Mathematics. 
 
 Chap. 9.—Exordial Address to the Student. 
 
 THE FIRST BOOK OF EUCLID. 
 
 Sect. T.—Definitions. 
 
 Sect. 2.—Postulates. 
 
 Sect. 3.—Axioms. 
 
 Sect. 4.—Definition of Terms. 
 
 Sect. 5.—Explanations. 
 
 Sect. G.—Preliminary Directions. 
 
 Sect. 7-—Problems and Theorems. 
 
 Sect. 8.—Exercises. 
 
 Sect. 9.—Supplementary Illustrations of Geometrical Logic. 
 
 USES OF EUCLID. 
 
 Chap. 1.—The Practical Uses of tho Definitions. 
 
 Chap. 2.—Miscellaneous Definitions, introductory to the Uses 
 of the Propositions. 
 
 Chap. 3.—Uses of the Problems and Theorems of Euclid. 
 
 fi^eAC 
 
 

 
 
 % 
 
 
 
 
M* r if/i /ho /' I h\ Lunik UUc X„J 
 
 > s INTRODUCTION TO THE 
 
 m/ 
 
 'JL fm/- 
 
 CHAPTER I. 
 
 
 § l B t COOUTlAOIKO - ^ P T. Trr j'iCR r . r . T iX A . I\ -. J I MHtAR 
 
 -y * rtVF.ll TlT^lMHim-IT PIT -I .IT l 
 
 in rann cTT tiHi . 
 
 «■ 
 
 st 
 
 ‘Whatever is worth wink; doing at all, it is worth 
 ■whale doing well/ is a maxim of deserved reputation: 
 and when applied in its proper sense, of great service. 
 
 But; a dop te d an . tin .. - ha s- bo a iq without ' dno ' y e gniul i o 
 
 ito tmnet impoit) it i we y he quest i oned whothop any 
 
 TY.Q V ;r» rn n in tl>n 
 
 mnjm-Hy 0 f \VhatTt is worth ve hilato d(^at all, moj $8 
 
 it is, unquestionably, sound wisdom and true economy } 
 
 to do well. But what it is worth wiiile-taqaerfe i tm , 8 'doiiutj 
 and the measure of the undertaking, are to be decided ’ J ‘ 
 
 ikj- 
 
 by the wants and circumstances of the person engag¬ 
 ing in the pursuit. Instead of this being done.we ;/ 
 have those who, on feeling interested in the investi- ' 
 gation of any branch of learning, conclude it to be 
 incumbent on them to entirely explore it, whatever be 
 its nature or extent, and, wanting time to accomplish 
 this, often an extravagant task, remain in entire 
 ignorance of that of which it was probably important 
 they should know .something. 
 
 The person destined for the profession of Chemistry, 
 (of neccsiitytjnus t^acquaint himself with the works of 
 I griartlnjr, Duty, Bal t or Ty j/idj rg , Faraday, Danie l s , 
 and oth e r eminent men in that department of know¬ 
 ledge : but the general inquirer will sufficiently in- 
 
8 
 
 IXTRODL'CTION TO THE 
 
 form himself on this subject from Chambers’ ‘ Rudi¬ 
 ments ’ of the science. In like manner, a person 
 intended for the Medical profession has t» wide olnaa 
 
 
 
 *st authors h i m , with whom he must become 
 
 conversant, while the artizan would be eon e idorod well 
 informed who was familiar with the works of Sout h- 
 wood Smith end Andrew Combey\Thow , ' r vhat i t' 
 pjymer for the Chemist and Medical student tojmtfer- 
 take^is no less than the whole range ofjstfowledge 
 belongmjmto each of their respective professions—and 
 this it is wb^li their while to m^sttfr well. But the 
 general inqiureSt gwi -t h o artizanToeing differently cir¬ 
 cumstanced, requircXmh^Such a general insight mto 
 these subjects as to converse thereon 
 
 with propriety^to understar^tkAooks in which they 
 are introduced, and to apply tnN^hiin and popular 
 rules Jkrfuic business, enjoyment, andp! 
 
 
 The rule of acquisition is this. After mas¬ 
 
 tering the knowledge proper tor individual profession, 
 we next have to seek general knowledge: for he 
 ^/"who should (timer to sound the depths of every sub¬ 
 ject, would find life exhausted before he had com¬ 
 pleted a tithe of his task. X oouw t .' illicit - woul d 
 umdomn to the imicmiLUirg toil uf hoaiding info i- 
 
 mat.ion flri/l f>vf ‘ lnr1n fni1 1,11 1 . ^'n 
 
 jduasun; of 
 
 ill ipplinntinm _ 
 
 Unmindful of eo T isidcialiunTiif this kind , Jt is com¬ 
 monly thought, that whenever Geometry is proposed 
 for investigation, (since whatever is worth while 
 doing at all is worth white doing well), nothing less 
 
 than Dr. Simpson’s Eight Books of Euclid, his name¬ 
 sake’s Trigonometry, Bonnycas^’s Mensuration. 
 Bridge’s Algebra,^ and a host of 
 must necessarily be a ntorod upo n. 
 
 the time of professej 
 the-Urpy(£i />/ 
 
 treatises, 
 tpen. it is only 
 
 students that wi >w of this, the 
 
 jfrC 
 
 'Tc^tCf f 
 
 / 
 
 (U*A. 
 
 
MATHEMATICS. 
 
 9 
 
 entire subject is generally neglected.* In what be 
 longs to a man’s profession, Ins duty, or his chosen 
 stucly, there must be no superficialness. 5 Fhc 
 j tMtvt ' ul - uotronomu 1 mu;l bo HUlMtT of NcwlOlb^-‘fra 
 mariner, of Thomas Simpson—the architgeT^nd civil 
 engineer, of Euler and the Bernomjiisfand a certain 
 class of able geometricians. Sp^mtich of geometry as 
 may belong to any branchj>fTfiechanics, must be mas¬ 
 tered by all who wiutftTbe proficients in such branch 
 —whatever bg^-fcfctffunimldblu array "of uuiks to-be 
 
 mil irrK'^ But under o t her oii'cumabme e * , what 
 
 is want eclats sucli a geneMl ueq u a i r iTance" w itTT 
 
 the subject as sharl enable a person to distinctly under¬ 
 stand the nature and application of Mathematics— the 
 process of geometrical reasoning—the meaning of the 
 technical terms now so frequent in the scientific lec¬ 
 ture room, and in treatises on Mechanics.! 
 
 * Of forty artizans who commenced the study of Mathe¬ 
 matics, under thtC-impression_in the Birmingham Mechanics’ 
 Institute (1837-9) scarcely o/w-eighth pursued tho subject 
 to a satisfactory issue: and tho few who did, pursuing 
 it at the usual length, had/to make large abstractions from 
 the hours of recreation ana rest, and, to a great extent, paid 
 their health as the penalty of their application. 
 
 t Greek and Latin, Aristotle's logic and classical versifica¬ 
 tion. quadratic equations/conic sections, the differential calcu¬ 
 lus, arc very good things/—They arc essential to tho fame of a 
 Parr or a Porson, a Herschel or a Whewell. These acquisi¬ 
 tions are, doubtless, amongst the greatest triumphs of tho 
 human understanding and are calculated to raise a few—per- 
 d—to distinction ; but upon the minds 
 ninety-nine they produce no sort of impres- 
 do much mischief to such persons in 
 r arc of incalculable detriment, and tho 
 absorb to no available purpose. Ten 
 years of youth—the most valuable and important period of 
 life—are wasted/m studies vrhich, to nineteen-twentieths of 
 the persons engaged in them, are of no use whatever in 
 future years. 7 ^-Blackwood's Magazine , Nov. 1845. 
 
 haps one in a hund 
 of the remaining 
 sion. They do i 
 themselves, but t 
 industry which t 
 
 cy 
 
 f ix. t k & 
 
10 
 
 INTRODUCTION' TO THE 
 
 Much lias been said of the qualities of calmness, 
 of clearness, coherency, and cogency—the reputed ^ 
 traits of a geometrical mind : and a general de¬ 
 sire of acquaintance with the writings of Euclid is 
 felt on this account. Wm , These points wminmw] jj 
 may be learned in a very satisfactory manner from 
 the First Book of Euclid, which six weeks’ application 
 will master. Un tins account tho - First Book only -ii 
 inlrnd i iml i n tliiw Work - -that Book- constitu t ing 
 
 a . complete in troduction tu ^ u aillL'ti ' icul ' scienc e - . 
 
 The usual works on ‘ Geometry/ and those under the 
 name of ‘ Euclid’s Elements/ have been mostly intended 
 for professional persons, or teachers, as few besidet 
 were likely to buy them. Sometimes they have beer 
 issued by individuals who merely sought geometrical 
 reputation. In both cases such works would natur¬ 
 ally contain much which the general student would 
 never want. Now that enqu i res diavo multi p li e d , 
 
 in the bustle ot 
 broken through 
 
 il ni podim t ry of the schools has been brushed lway 
 practical life, and zeal of lea rning 
 the barriers of caste — the 1 opu- 
 
 G e o me tr y . Euclid, by the unanimous consent ol 
 learned men, is the keeper of the gate* His First 
 Book is the entrance to the Temple a t m it io strange 
 
 th a t b wii ) among the moderns haM^buliiULd this v u U i- 
 
 l mlo -for iho admicoio n- 
 
 WHaewm -o f tho imaot oe h 
 
 last 
 
 -imy u 
 
 he m 
 
 public tr r v iew the 
 
 Proclus, the lasLptfeek geometer, seems to have 
 had this idea. Hualone, of the ancients, is recorded 
 to have paid distinguished attention to the First Book 
 of Euclid./He left a special commentary upon it, 
 abounding in curious observations on the metaphysics . 
 of EjuSnd, and with curious facts connected with th efull 
 sconce. 
 
 The desirability of recasting our educational trea- 
 
MATHEMATICS. 
 
 11 
 
 tiscs the commercial habits of the age leave be¬ 
 yond doubt. The industrious classes are thjt/who 
 most need the acquisition of any logical power which 
 Geometry can impart, and their means of leisure to 
 pursue such a study are few and far between. Ac¬ 
 tuated, probably, by convictions of this kind, Sir John 
 Herschel lias composed a work on Astronomy, in¬ 
 tended to answer for the astronomical, such purposes as 
 are here contemplated for t h* geometr y *!- publ io . 
 Indnndj lurry ■■■■ -- i ■ ^ l^.m, Kimn.i '■ '• 
 
 at tention to tho onme objootw the ir- respective rriniry.; . 
 
 O f the " ' fi t ness of the " si ng le b oo k -o f - Eucl i d for th e 
 
 fr f H i ' ■ ■■■1 . 1 . 
 
 rntirn ft pn^f " Since the writer first arranged 
 thorn; a coincident opinion has been very distinctly 
 expressed upon the subject.— 
 
 4 To young readers, about to enter on the study of 
 Astronomy, and who seek only to get such a mere 
 general knowledge of it as may satisfy their own 
 minds, we would say in the first place, acquire a 
 knowledge, however slight it may be, of the elements 
 of Mathematics. Your mind may not be of a mathe¬ 
 matical turn, and there may not be the slightest pros¬ 
 pect of your deriving any positive advantage from 
 posing your brains with the “ First Booh of. EilcImU' 
 No matter; try and go over it; it is worth your 
 while. You cannot stir in Astronomy without know¬ 
 ing something of the properties of the circle and tri¬ 
 angle. He, therefore, who wishes to comprehend the 
 c< reasons ” on which Astronomy is based, will acquire 
 this preliminary knowledge, without which it is useless 
 for him to enter upon the study. After he has ac¬ 
 quired it, and after he has studied an Astronomical 
 work, he may be far—very far, indeed, from having 
 the smallest pretension to the name of Astronomer, 
 
 £>J>e 
 
 
 W 
 

 12 
 
 INTRODUCTION, LTC. 
 
 
 but he will be in possession of a few of the “ funda¬ 
 mentals ” of the science; and will stand on the same 
 platform with the Astronomer himself.’* 
 
 The world has been favoured with the Beauties of 
 , of Byron, Scott, ami of * to g r eat bcnefaetor» a 
 li tera t ur e and f»ng, awl the pubho who ooul 4 ~ w ov e r 
 
 acfjunint them s el r es tr ith (lie I ’TTtr nr piuduaiciu-o f 
 
 t h r fr tnithorcj ha r e Imc p mu - familiai, th r ough STr cfr c o n- 
 
 . veniant ap i tome.^ wi t h- t he ir me r it s and thei r graces . 
 
 c*1 Uuire . arft . hnth romtmo fr- &»d beauty of a stern kind 
 
 in the wonderful writings of Euclid, no ra a n - of i nfbr - 
 H ttrf i o tt- or t aste w il l deri v e His works were the torch 
 which lighted the sciences to their earliest discoveries. 
 They revealed t o- the world of form and the 
 hidden secrets of figure, and enlisted lines and 
 angles in the eternal service of man. The ' deep -on- 
 t 4 »u:;ia s n r- ■vedne h 7 t)inu liuvu ful t fur Qumnel i y —mil 
 
 bo -gonowdh 
 
 2 
 
 mum 
 
 ohared ■ 
 
 in — th e- curren t—d iffuci ea—of 
 
 i nlol li gowa e. A h ■ impo rt an t connection is beginning 
 to be established between the principles of science and 
 morality. More than one philoso^r is found to ac- £7) 
 knowledge, that ‘The axioms of Physics translate ' 
 those of Ethics.’f The desire of aiding to some extent 
 the development of this fact was not one of the least 
 reasons which has dictated the compilation of the 
 ‘ Beauties and Uses of Euclid.’ 
 
 * London Saturday Journal, 
 j Emerson. 
 
 A 
 
CHAPTER II. 
 
 frMPirnTATf T DISTINCTIONS PERTAINING TO 
 MATHEMATICS. 
 
 Mathematics is the science of quantity and magni¬ 
 tude. It is usually defined as that which treats oi 
 whatever can be numbered or measured.* Mathe¬ 
 matics embraces all modes of procedure, and includes 
 all the arts and sciences employed in determining 
 magnitude and quantity. Quantity is understood to 
 include the distinctions of weight and measure. Mag¬ 
 nitude is the consideration of size. 
 
 Geometry , though a very important science, is but 
 a branch of the Mathematics. Geometry treats of 
 the properties of lines, angles, surfaces and solids. 
 The term Geometry is applied to Mensuration, to 
 Trigonometry, Conic Sections, and to every depart¬ 
 ment of investigation in which lines and angles, sur¬ 
 faces and solids, are chiefly concerned. 
 
 Arithmetic, the art of figures — and Algebra, 
 the science of symbols, constituting the lower and 
 higher departments of calculation, are both branches 
 of Mathematics. But Geometry is the principal 
 branch. 
 
 ■ of' definition t he nululi r e dia ¬ 
 mond the branched of Mathe- 
 •erceived, and the careful obser- 
 avoid the error of confounding 
 le common mysterylattached to 
 in a great measure^ its • rise in 
 r. 
 
 * Rhine’s Principles of Geometry, page 2. 
 
14 
 
 INTRODUCTION' TO TIIE 
 
 
 Ci_ 
 
 ? k , il . led in the use of figures is an Arithmo- 
 
 dcsiguatX ,£ 
 
 aSertL, C '° nt “ E “ cUd ’ s Elcmenl » is a 
 
 £?.”? ,u a 'V,, 1 . ll ° morc . P ro clse ancients always 
 
 Gr . cat Gaamcter. It Ts ^person wholas « 
 quired skill m all thes^iranches, rmd-imeh uu'tu/ns. 
 
 proi,oriy - thc Ma ‘ h - a - 
 
 W«» often, mdoofl lmrtr the H-ilf.,! Dh .y 
 
 a Matl ematieian. The Tpicmnn»««*^«* ^JL?m atylui 
 
 trician 
 correct 
 terms. 
 
 . Tlic Trigonometrist 0 al<f ~Gcome- 
 
 often receive the same designation. But 
 taste is never guilty of this infusion of 
 
 i U ;±”“!l f Z” 9ion is a Poor Compliment 
 
 _ . . 
 
 to the ] recisest of sciences. 
 
 nM ‘ J Matliomfttieo w i ll never be 
 
 sense ks explained in its definition. That term on 
 the till* page is employed to indicate tha ; an attempt 
 Y • ljC V made . to „ la y l>are its clear impo t, and by so 
 °a lts . re i 3ut ? d mystery; ,nd illustrat- 
 ng (as Will bo done m various places) tie exact na¬ 
 ture an} influence of its branches, and a ford a com¬ 
 prehensive and easy insight mto its nature and applica¬ 
 tion. Efut chiefly, this work, as its sec >nd title ex- 
 
 p^sses, ISTUiilinul tu Euclid—to Ccomcttr. - 
 
 The Elements of Euclid are often designated the 
 Elements of Mathematics. Ti rp Qfhc Elements once 
 constituted Mathematics themselves; but since the 
 perfection of numerals and thc invention of Algebra 
 by Diophantus, they have sunk into a relative rank. 
 It is true, Geometry gives laws to Arithmetic. The 
 
MATHEMATICS. 
 
 15 
 
 Second Boot of Euclid powers ol 
 
 numbers. None of the ancients unfolded v 
 
 r- UT1 d theory of proportion with uuoh ubilrty as Euclid^ 
 Shta Fifth BooE. P Thel6thProposidonof r ^S^ 
 Book is the foundation of the Rule of Three. 1 
 e£ Euclid is, to a great extent, the logic of thej^dtlie 
 m^g^But though these cardinal and cpnrmon qua- 
 EsmSU make the First BpS^uitroduction 
 
 to Mathema^^^we n^tSnTueh war¬ 
 
 ranted in confounding>^for a momen^ Ae l^ser 
 
 with the grcaten^Sc Elcm&i^ofthe Mathematics 
 signify thc^eSients of all the branche&^ft^e Matlie- 
 rnaticarrhe Elements of Euclid are thei*«miils of 
 gometrv. 5 
 
 ■ COIllft 
 
 - duatme w ona, cauaca^vui^ —- — -- 
 
 L jM*uU of Euclid, put the mU» question Is a know- 
 of arithmetic requisite before commencing the 
 stu3y of the Elements V I t wowiy feet qiiu i :ti o n,Hw l 
 U - has been the fii'ot qacatiwof-wy pupili t o mr . Iso 
 wuiuii 'T c o uld - bo ■ moi ' fr i od fo i v If the pupil was 
 enterin'* 1 on the study of the entire Mathematics, it is 
 plain he wants no arithmetic to begin with, ati t‘“ 
 pi iui' in ii : n f M n thrnn i li ■ " ■ !.. t- bui irrt. -~Ar»4 G eo- 
 
 me try that teaches t\\e.science\ of numbers. irs~ 
 
 - A ~r - — --— 
 
 ie Books of Eucl d aro a scries of graduated steps, 
 and wh\n -we have lear icd the principle of Ynounting the 
 ladder can climb to aty height. The First BAok deyelopcs 
 these principles. It is In this sense that the First Book is 
 here put Spr the whole fif een. 
 
 + In another place (Pr .ctical Grammar, p, 9.) A have found 
 it necessair to explain \ hat it may be useful to Repeat here, 
 that ‘ an Ait is a system of rules, for the performance of any 
 operation—and Science i 5 that which explains the Reasons on 
 which the rules of art are founded.’ The Elements o\ Euclid 
 being eminently a demonstrative science perform th\s office 
 
 U 
 
 L 
 
16 
 
 INTRODUCTION TO THE 
 
 .age nts ail) limp nnil upglue.— It / io u diothie > , ent ire, 
 and independett science. 1 
 
 But it will be asked by the/intelligent leader, if 
 Mathematics Comprises not only/Arithmetic,/Algebra, 
 Differential and Integral Calculus, and Gebmetry— 
 but also such principles as are employed in the 
 measurement bf the earth and the skies, inf Physical 
 Geography, /Astronomy, Navigation, Architecture, 
 and a variety of other branches, how is the maxim 
 with which pec. I. opens to be applied by /the general 
 enquirer to these departments? The answer is easy. 
 
 If the enJquirer aims at the eminent distinction "of 
 being a M/ithematician, he must prosedute each of 
 these branches of study, so ‘as to acqui/e familiarity 
 with their/modes of procedure, and some skill in their 
 application. Nor will this be so difficult as at first 
 sight appears. The nature^of each islkindred, and 
 Geome/ry is the key which ii ill unlock them all—only 
 it requires the application or many years to compass 
 so wide a range of physical*'knowledge./ 
 
 The course of the gerieral enquirer is different. 
 Arithmetic being required, to some extent, by every- 
 
 for Arithmetic and several other arts. T^io Arithmetician 
 and practical Geometer, ^'distinction hereafter to be ex¬ 
 plained) come to Euclid for the reasons of tnteir rules. I am 
 aware mat Schiller (according to the Dublin University Maga¬ 
 zine ) makes the great distinction between art and pure science 
 to consist in the elementfof humanity beingnlways essen¬ 
 tially inAdved in every work of art, and this (dement being 
 excluded m all that is projjerly called science, llut this dis¬ 
 tinction mates to high ^rt—to the poetry of Wt. I have 
 here to do with art in tho sense of manipulations— ahd^wfth 
 science in tne sense of systematized and demonstrluive know¬ 
 ledge—and m every employment of these terms, allusion is 
 made to thc\art and sdienco of practice. No (\isparage- 
 ment to Schilfbr’s distinction, but ours is the othej depart¬ 
 ment. f 
 
MATHEMATICS. 
 
 17 
 
 body, he acquires .itk first principles, and as much oi 
 its practice as' Ke'isTikdy td"TTCCd'.' '"If’ lie is curious 
 about Algebra, he may’ soon gather j from Coles’s 
 admirable ‘ Conversions’ on that subject, the requisite 
 information—the Fi|sl Book of Euclid will accomplish 
 the rest. It will introduce him to both Theoretical 
 and Practical GeometW, and to the technical charac¬ 
 teristics of Mathematics! It will enable him to under¬ 
 stand the modus ojwraMi of Newton’s Principia. It 
 will familiarise him with the illustrations of Dynamics 
 and Mechanics. After berforming the ‘ Constructions 
 of Euclid, and putting them to the ‘ Uses ’ at the end 
 of this book, any person teould understand and practice 
 Keith’s principles of Mapping, and illustrate his 
 geography by constructing his own maps. We have 
 seen in the quotation iA the last chapter, from the 
 Saturday Journal , how tthe First Book of Euclid, il 
 it does not make the youna pupil master of Astronomy, 
 it will place him on the lame common ground with 
 the Astronomer himself, abd familiarise him with the 
 scientific secrets of that fine pursuit. Indeed, if the 
 First Book of Euclid does not make the student a 
 Mathematician, it so qualifies him that he can be a 
 Mathematician whenever lie pleases, as it imparts a 
 clear insight of that science which will enable him to 
 extend his studies whenever his duty or taste shall 
 incline him. Thus the genital enquirer applies our 
 maxim. He determines wllat is properly worth his 
 attention, and earnestly devites himself to that, and 
 to nothing else. His business is to explore , not to 
 conquer the region of Mathematics It is that which 
 i t .ip ■worth his while ttr du, ~ that well .—*— 
 
 Man y people of undisciplined powers, and ignorant 
 of the art of acquiring knowledge, can never move 
 anywhere unless they move profoundly. Such people 
 soon Lose themselves. Instead eg being the masters of 
 
18 
 
 INTRODUCTION TO THE 
 
 knowledge, knowledge is the master of them. By 
 
 Umg - pvofomid o nt of pIao a ; th»y bri n g profund ity 
 
 w«o contempt:— It 1 ir-nut w i t hin tho oampaca of huma n 
 p owe r ■to he [ ' nufoi iTi th ' upiim moro than - a few s ubjoota 
 
 The poet was somewhat too strict in his limitation, 
 but he approached the truth in declaring— 
 
 Ono subject only will one genius fit ; 
 
 So vast is art—so narrow human wit. 
 
 Do not be deterred from the useful course of general 
 investigation, in those two Sections recommended, by 
 the v&guo charge 'of-superficiality, so often preferred 
 without propriety and listened to without advantage. 
 Many persons live and die in disreputable ignorance 
 because they are determined not to be superficial. 
 Superficiality is a serious imputation when it respects 
 what you ought to know well. But with respect to 
 general knowledge, it is no more criminal to be super¬ 
 ficial than to have only five senses, or not to be in half 
 a dozen places at once. It is the condition of humanity 
 only to be able to take a general survey of the wide 
 field of knowledge. Thq republic of literature has no 
 inapt resemblance to the state of good citizenship, in 
 which a man is expected to know his own business, 
 and to have such an acquaintance only with that of 
 his neighbours as shall give him a laudable interest in 
 their pursuits, and assure him that they are conducive 
 to the dignity of the comjnonwealth. 
 
 - ^X Tinr mijori^ uf the br anches ot kiulil l lg, we ca n- 
 
 <)idyhr>ptiio master the first principles^^Arrrdu 
 happens tjiar^a^work• that lias ever 
 
 devised teaches thelrrtj^S^ffgtliis in so striking 
 a manner as does,^th<f^^st^I?fiok^^'Euchd. Byt 
 more of tais>irtTie proper place. W^sujst now in- 
 trodus^tto^rcader to the History of the Matl!e«a|itics, 
 which he is prepared. The record • of its gfa- 
 
mathematics. 
 
 19 
 
 dual risk out of the wants of men by patient study, 
 and its progress, will further divest him of the preju¬ 
 dice, if my remain, of the popular mystery connected 
 with thislscience. 
 
 CHAPTER III. 
 
 HISTORY OF 
 
 LA-n pnnnnjyia 
 
 MATHEMATICS. 
 
 ‘Thu Mathematical Sciences were the first of all 
 among men, if we may believe Josephus. He (Bk. I. 
 Chap. 3,) writeth, that the posterity of Seth observed 
 the order of the Heavens, and the courses of the Stars. 
 And lest these inventions should slip out of the know¬ 
 ledge of men, Adam having predicted a two-fold des¬ 
 truction of the Earth, one by a Deluge, and the other 
 by Fire, they raised two Columns, one of Bricks, the 
 other of Stone, and inscribed their inventions upon 
 them ; and if the Brick one should happen to be 
 destroyed by the Deluge, that of Stone, which would 
 remain, might afford men an opportunity of being 
 instructed, and present to their view the things which 
 had been inscribed upon it. They also say, that that 
 
 • Compiled chiefly from Andrew Tacquet’s (of the ‘ Society 
 of Jesus’) Elements of Euclid, translated by Whiston—the 
 Introduction to the Elements of Geometry, translated into 
 English from Peyzard, by Phillips—and from the History of 
 
 flip fieienee nrefixed tin the Elements of Plane fienme*.... T... 
 
 Bell. JLiut, 
 
 inventions 
 
 of course, omitting’ iliu usually rue 
 
 
 who, on a 
 enraptured 
 Loci,’ Pori 
 duced when 
 
 general introduction to Mathemat 
 or be enlightened by details of * r I 
 Quadratics, and Curves ? They 
 they are intelligible. 
 
 ics, would feel 
 he Lair,’ ‘ The 
 are best intro- 
 
INTRODUCTION TO THE 
 
 20 
 
 Stone Pillar, which even in our day is seen in Syria, 
 was dedicated by them. This Josephus says, whom 
 I leave to vouch for the story/* 
 
 Geometry must be coeval in its origin with the in¬ 
 vention of the square and the compasses, and these 
 being necessary to the rudest operations, were doubt¬ 
 less known in the earliest ages of the world. Geometry 
 first assumed the character of a science, in Greece, 
 about 2,500 years ago, and flourished during ten 
 centuries: then was almost entirely neglected till 
 about 200 years since, when it broke forth in Europe 
 with more than its pristine brilliancy. 
 
 The term Geometry is derived from the Greek 
 words ge (earth) and metron (measurement), literally 
 signifying the Measurement of the Earth. But the 
 science thus designated is now so extended, as to be 
 equally applicable to the measurement of the heavens. 
 Aristotle, the famous preceptor of Alexander the 
 Great, ascribes its origin to the Egyptian Priests, 
 and Herodotus, an ancient historian, to the time when 
 Sesostris, who reigned in Egypt, (1500 B. C.)f inter¬ 
 sected it by numerous canals, and apportioned the 
 country among the inhabitants. 
 
 To Egypt appears chiefly due the honour of having 
 been, if not the birth place, at least the cradle of Geo¬ 
 metry, where it received its first culture. That old 
 and mighty kingdom seems to have been the nurse, 
 perhaps the mother, of almost all the useful arts and 
 sciences. Ilindoostan and China contend with Egypt 
 for the pre-eminence with respect to Geometry ; but 
 by whichever it may have been first cultivated, it 
 
 * Tacquet’s Historical Account of the Rise and Progress 
 of Mathematics. 
 
 t B. C. Before the reputed Birth of Christ—before the 
 lominencement of what is denominated the Christian Era. 
 
MATHEMATICS. 
 
 •21 
 
 was certainly brought into general notice by the Egyp¬ 
 tians. 
 
 Thales, the reputed father of Greek philosophy, born 
 at Miletum (640 B. C.), is celebrated for being 
 the first who introduced Geometry to Greece, and 
 for the many highly interesting discoveries he made 
 in it: for being able, by geometrical principles, to 
 compute the distance of vessels from the shore, and 
 for having measured* the altitude of the Pyramids of 
 Egypt by means of their shadow, much to the as¬ 
 tonishment of Amasis the king, who wondered at his 
 scientific skill. Prom him Astronomy is said to have 
 made considerable advances—and lie founded an estab¬ 
 lishment in Greece known by the name of the Ionian 
 School. It was in the age of this Philosopher, accord¬ 
 ing to the best informants, that Geometry first began 
 to assume the character of a science. He observed 
 the Equinoxes and Solstices, and was the first who 
 foretold the Eclipse of the Sun. And he foretold to 
 King Cyrus an Eclipse of the Moon. The Electrical 
 properties of amber were first remarked by him, upon 
 which has been built the important science of Elec¬ 
 tricity. 
 
 The first elementary treatise on Geometry is said 
 to have been composed by Anaximander, a disciple ot 
 Thales, and Thales’s successor in his school. 
 
 Next in order comes Pythagoras,! the philosopher 
 
 * Thales, one of the seven wise men of Greece, about (J00 
 years before Christ, invented the following method for mea¬ 
 suring the height of an Egyptian pyramid. lie watched tho 
 
 { >rogress of the sun till his body and shadow were of the same 
 ength, and at that instant measured the shadow of tho 
 pyramid, which consequently gave its height.— Lord Karnes 
 JJ is tori/ of Man, vol. 1, p 91. 
 
 t The supposed inventor of the term ‘ Philosophy ’—sig¬ 
 nifying a lover of knowledge. A term said to have been first 
 applied to Solon, an ancient law-giver. 
 
22 
 
 INTRODUCTION TO THE 
 
 of Samos, (an island in the Levant, between Greece 
 and Asia), no less distinguished than Thales for the 
 variety and extent of his discoveries. He is said to 
 have travelled into Egypt and India in pursuit of 
 knowledge, to have settled at Tarentum in Italy, 
 (about 550 years B.C.), and to have maintained, in 
 Astronomy, the true system of the world which places 
 the sun in the centre with the planets revolving 
 round it. From this philosopher it was called the 
 Pythagorean System, and was revived two thousand 
 years after his day by Copernicus. Pythagoras was 
 distinguished in all his pursuits by a genius remark¬ 
 ably inventive and assiduous. He possesses no ordi¬ 
 nary claim to the honour of posterity. As a mathe¬ 
 matician, he v r as decidedly the first of his time,—as a 
 philosopher, he delivered many curious things con¬ 
 cerning God and the human soul, and a great variety 
 of precepts relating to the conduct of life, both civil 
 and political. 
 
 It is not certain at this time whether Geometry had 
 been founded into a regular system. If not, consider¬ 
 able advances had been made, and ere a century had 
 elapsed from the age of Pythagoras, Zendorus, a man 
 of great parts, arose, whose writings are the first 
 among the ancients which have survived the w^reck of 
 time, one geometrical tract of his having been pre¬ 
 served. 
 
 Then came Hippocrates, whose brilliant genius, 
 great diligence, and application, rendered essential 
 service to this science. lie was the third who wrote 
 on the Elements of Geometry, but on the appearance 
 of the Elements of Euclid, his treatise was consigned 
 to oblivion. 
 
 The origin of the celebrated Platonic School, (400 
 years B.C.), is considered one of the most important 
 epochs in the history of the science. Plato, its 
 
MATHEMATICS. 
 
 23 
 
 founder, who is called by way of pre-eminence the 
 ‘ Athenian Sage/ and who was alike remarkable for 
 eloquence, the poetical sublimity of his imagination, 
 and the accuracy of his mathematical performances, 
 gave to Geometry the form and substance of a com¬ 
 plete science, and enriched it with many discoveries. 
 Indeed, so profound a veneration did he entertain for 
 it, that he made it a principal object of instruction 
 among his scholars, and had written over the entrance 
 of his academy, the well known words ,—‘ Let no one 
 enter here who is ignorant of Geometry .’ Plato was 
 the discoverer of the Conic Sections, (or those curves 
 which are found on the surface of a Cone when cut 
 through in different directions). He found out many 
 of their most remarkable properties, which being 
 made the continual study oi his scholars and succes¬ 
 sors, at length became a distinct science from the 
 common elements, and received the appellation of the 
 higher or sublime Geometry. 
 
 Such was the versatility of his genius, that he 
 directed his acute and original powers with eminent 
 success to the cultivation of Moral Philosophy, and 
 produced his famous work, entitled the ‘ Republic/ 
 with the view of imparting to civilised intercourse 
 the character of a science, and giving a philosophical 
 direction to the ever jarring aims and interests of 
 mankind. 
 
 The celebrated Aristotle, the successor of Plato, 
 and preceptor to Alexander the Great, and who is 
 justly regarded as one of the greatest men of olden 
 time, made many improvements in the mathematical 
 sciences. After attending the lectures of Plato, he 
 opened a school himself in the Lyceum of Macedonia, 
 which was assigned him by the magistrates, and was 
 the founder (about 350 years B.C.), of the well known 
 Peripatetic school. 
 
24 
 
 INTRODUCTION TO THE 
 
 Theophrastus, a disciple of Aristotle, composed a 
 complete history of the origin and progress of Mathe¬ 
 matics, Astronomy, and Arithmetic, from the earliest 
 periods to his own time. The treatise has been unfor¬ 
 tunately lost. 
 
 Archytas, of the academy of Plato and of Taren- 
 tum, the place in which Pythagoras settled, is famous 
 as the constructor of a flying pigeon, and as the re¬ 
 puted inventor of the pulley and screw. ‘ He was one 
 of the first/ says Tacquet, * who brought down the 
 Mathematics to human uses.’ Aristceus, who flour¬ 
 ished (about 300 years B.C.), wrote many works of 
 considerable merit, acquired great proficiency in 
 Sublime Geometry', and is said to have been the in¬ 
 structor of Euclid. 
 
 Euclid, according to Pappus and Proclus,was born at 
 Alexandria,* in Egypt, where he flourished and taught 
 Mathematics with great applause, under the reign of 
 Ptolemy Lagus, (about 280 years B.C.) Some Ara¬ 
 bian historians, however, inform us, that he was born 
 at Tyre, that his father’s name was Nancrates, an in¬ 
 habitant of Damas. The particular place of his na¬ 
 tivity appears, therefore, uncertain, but whether or 
 not Alexandria had the honour of giving him birth, 
 all historians agree that he flourished there as a 
 teacher at the time above mentioned—which city, from 
 that period till the time of its conquest by the Saracens, 
 seems to have been the residence, if not the birth place, 
 of all the most eminent mathematicians. It is sup¬ 
 posed that Euclid studied at one time under the disci¬ 
 ples of Plato at Athens. History is silent as to the 
 time of his death. Pappus represents him as a per¬ 
 son of great courtesy, mildness, modesty, and benevo- 
 
 * A town built by Alexander the Great, near the mouth of 
 the Nile, in Egypt. 
 
MATHEMATICS. 
 
 25 
 
 lence. He left various profound treatises on the most 
 abstruse subjects belonging to his profession, but 
 his eminent work is his Elements of Geometry, 
 which supplanted all of a similar kind that preceded 
 it, and such was the judgment he displayed in its 
 composition, that notwithstanding the great additions 
 made to Geometry since his time, it has continued for 
 upwards of 2000 years to sustain the highest reputa¬ 
 tion as an elementary treatise.* 
 
 The great reputation of Euclid arose from the ex¬ 
 cellence of his Elements. His merit in this perform¬ 
 ance consisted in his having systematised the scattered 
 discoveries of Thales, Pythagoras, Plato, and the 
 lesser geometers who preceded him. No doubt he 
 supplied important links. But when he infused co¬ 
 herence into the fugitive inventions of his predeces¬ 
 sors, he imparted to Geometry the importance of a 
 science. 
 
 Next after Euclid came Archimedes, who was born 
 in the Island of Sicily. He established the true 
 principles of mechanics, laid the foundation of Hydro¬ 
 statics,+ traced out the first rudiments of Naval 
 Architecture, and applied his talents with such 
 success to the cultivation of Geometry, Arithmetic, 
 and Optics, that he has been styled the Newton of 
 antiquity. He is said to have prosecuted his studies 
 to the neglect of both food and sleep. I He perished 
 
 * The curious may be interested to learn, that tho 
 Elements of Euclid were first published in Latin, in folio, by 
 Radholt, at Venice, in 1482. First published in the original 
 Greek by Hervagius, folio, Basil, 1533. And the first English 
 translation was by Billingsley, folio, London, 1570, with a 
 curious preface by John Dee. 
 
 f The science of weighing fluids. 
 t No great proof this of his ‘ philosophy.’ Nothing is 
 philosophical which cannot with prudence be imitated. 
 
INTRODUCTION TO THE 
 
 26 
 
 at the fall of Syracuse. A soldier came suddenly 
 upon him in the museum, when he was intently en¬ 
 gaged. on some geometrical problem, and commanded 
 him to follow him to Marcellus, (the Roman Consul 
 who had taken the place). He refused, however, to stir 
 till he had finished the subject on which he was en- 
 gaged, and the soldier ran Archimedes through with 
 the sword. When dying he desired that a sphere in¬ 
 scribed in a cylinder might be engraved on his tomb 
 to perpetuate the memory of his most brilliant dis¬ 
 covery.* The Sicilians, however, soon forgot him, 
 and his tomb was found 200 years after by Cicero, 
 by means of the symbol just named, in a field near 
 Syracuse, overgrown with thorns. He was the most 
 inventive, and the greatest Mechanical Engineer of 
 ancient time. 
 
 Eratosthenes, a native of Cyrene, was called to the 
 Alexandrian school by Ptolemy Euergetes, who made 
 him his librarian on account of his extensive and 
 general attainments in learning. His knowledge of 
 Geography, Astronomy, and Geometry, enabled him 
 to discover, for the first time on record, a method of 
 measuring the circumference of the earth. 
 
 Fifty years after the death of Archimedes, Apollo¬ 
 nius appeared, who gave a great impulse to the ma¬ 
 thematical sciences, and whose discoveries were so 
 highly esteemed that he was honoured by the appel¬ 
 lation of the Great Geometer. 
 
 From the time of this eminent man we move on 
 for three or four hundred years without meeting with 
 one person who contributed to the advancement of the 
 sciences. 
 
 * He discovered some very fine proportions which exist be¬ 
 tween the sphere and the cylinder. 
 
MATHEMATICS. 
 
 27 
 
 Then came Theon, (about 380 years, A.D.*) who, 
 by his skill and perseverance in Mathematics and 
 Philosophy, obtained the honourable dignity of being 
 appointed president of the famous Alexandrian School, 
 where, by his condition and conduct, he gained the 
 greatest respect and reputation. 
 
 Theon had a remarkable pupil in his own daughter, 
 Hypatia—a lady learned as well as beautiful. She 
 was born at Alexandria. Hypatia was mistress of all 
 the ordinary accomplishments of her sex, familiar 
 with the most abstruse sciences, and made such 
 progress in Philosophy, Geometry, Astronomy, and 
 the Mathematics generally, that she was held as the 
 most learned person of her time. She published com¬ 
 mentaries on Apollonius’s Conics, on Diophantus’s 
 Arithmetic,f and other works. She was chosen whilst 
 very young to succeed her father in the same school, 
 and to discourse from the presidential chair, at a time 
 when Ammonius, Hierocles, and many other very 
 learned men abounded at Alexandria and in various 
 
 { >arts of the Roman Empire. Among her pupils, not 
 ess eminent than numerous, was the much esteemed 
 Synesius, afterwards Bishop of Ptolemais. But it was 
 not Synesius only and the disciples of the Alexandrian 
 School who admired Hypatia for her virtues and learn¬ 
 ing. She was held as an oracle by the public, and 
 was consulted by the magistrates on all important 
 occasions. In snort, when Nicephorus intended to 
 pass the highest compliment on Eudocia, he thought 
 ne could not do it better than by calling her 
 
 * (380 A.D.)—380 years after Christ: A.D. being the ab¬ 
 breviation of Anno Domini, the year of our Lord. 
 
 f ‘ Diophantus was as great in Arithmetic as Archimedes, 
 Apollonius, or Euclid was in Geometry, and by him was found 
 out that admirable art we call Algebra. 1 — Tucquet. 
 
 C 
 
23 
 
 INTRODUCTION) ETC. 
 
 another Hypatia. Whilst Hypatia thus reigned 
 the most brilliant ornament of her sex in the annals 
 of history, she was greatly admired by Orestes, 
 the governor of the city, who, on account of her 
 wisdom, often consulted her. This, together with an 
 aversion which St. Cyril had against Orestes, proved 
 the cause of her ruin. About 500 monks assembled, 
 attacked the governor one day, and would have killed 
 him had he not been rescued by the townsmen—and 
 the respect which Orestes had for Hypatia, causing 
 her to be traduced among the Christian multitude, they 
 dragged her from her chair, tore her in pieces, and 
 burned her limbs.* This shocking and brutal catas¬ 
 trophe was perpetrated in the lent of the year 416, in 
 the time of Theodosius II.+ 
 
 Pappus, a consummate mathematician, flourished 
 about the end of the 4th century. His work, entitled 
 ‘ Mathematical Collections,’ has transmitted his name 
 to posterity with distinguished lustre. 
 
 Proclus, who was at the head of the Platonic 
 School, established at Athens (A.D. 500,) rendered 
 important service to the sciences, and showed great 
 kindness to all who embraced their pursuit. 
 
 The age of important discovery was past before 
 the time of Pappus—and science in his time was 
 beginning to decline. The labours of Proclus and a 
 few of his followers were the last expiring efforts 
 
 * It is related by Damasius and Suidas, and clearly proved 
 by such learned men as Bruker, La Croze, and Basnaoe, that 
 St. Cyril incensed the Christian populace against her .—■{Vol¬ 
 taire's Phil. Diet. art. Hypatia.) St. Cyril was a man of parts, 
 and there is a difficulty in accounting for tho inhuman prin¬ 
 ciples that actuated him in this affair. 
 
 t For a more particular account of this illustrious victim 
 of fanaticism, see Bossut’s Hist. Mat., English Ed., 8vo., 1803. 
 
29 
 
 
 MATHEMATICS. 
 
 made in Greece for ancient Geometry. The Alexan¬ 
 drian School still existed, and science might have con¬ 
 tinued to flourish, had not the conquest of Egypt by 
 the Arabians, who, with the design of spreading their 
 religion and their empire, inundated Christendom in 
 the seventh century, and not only gave it a fatal blow 
 in Alexandria, but also extinguished its nms languid 
 existence in Greece. 
 
 Throughout all the civilised parts of the world in 
 those times, the Alexandrian Library was known as 
 the treasure-house of literature and knowledge. But 
 neither the members nor the materials of this sacred 
 edifice were respected by the Arabians, who, stimu¬ 
 lated perhaps more by religious fanaticism than bar¬ 
 barous ignorance, sacked Alexandria, and massacred 
 the philosophers who were assembled there for the 
 purposes of study from all the neighbouring countries. 
 Their laboratories and instruments were destroyed, and 
 finally the Library itself, the accumulated scientific trea¬ 
 sure of ages, containing above 200,000 written volumes, 
 was committed to the flames for the mean purpose of 
 heating the public baths. Caliph Omar, as he gave 
 orders for the annihilation of the books, justified it by 
 the strange, but convenient argument, that —‘ if the 
 volumes were conformable with the Koran, they were 
 superfluous; and if contrary to it, impious.’ With this 
 brutal act of bigotry and devastation may be said to 
 have closed the scientific career of antiquity. 
 
 As a partial amends for so odious an outrage against 
 civilization, the Arabians, about a century after, began 
 to cherish the sciences of Geometry and Astronomy, 
 and thus afforded an asylum which preserved them 
 from entire extinction. About the time of their 
 learned prince Almamon, several of their most cele¬ 
 brated works on Geometry were translated into the 
 Arabic. Geber Ben Aphla laid the foundation of the 
 
30 
 
 INTRODUCTION TO THE 
 
 modern Trigonometry. An elegant treatise on Men¬ 
 suration was written by Bagdadin; and it appears 
 from his treatise on Optics, that Alharin was a superior 
 geometer. They studied also the more sublime Geo¬ 
 metry, improved Trigonometry, and reduced it to a 
 convenient form. 
 
 Of the few Persian geometers, Nassir-Edin and 
 Maimon-Rescliid were the most distinguished. They 
 both composed commentaries on Euclid. Persian 
 learned men called Geometry the difficult science, 
 and did not make any improvements in it—neither 
 did the Turks nor Hebrews, although they translated 
 some of the elementary works. 
 
 In one of the ancient treatises on Astronomy of 
 the Hindoos, called Suryd Sidhanta, there is a correct 
 system of Trigonometry; and hence it is concluded 
 that the Indians had cultivated Geometry before the 
 composition of this treatise. The Chinese were ac¬ 
 quainted with the 47th of the First Book of Euclid. 
 
 The Romans knew little more of Geometry than 
 the practice of land measuring. Varro, however, 
 composed a treatise on Geometry, and Cicero had 
 some acquaintance and esteem for it too. Vi¬ 
 truvius and Boetius had some knowledge of Geo¬ 
 metry, but the latter is better known for his ‘ Conso¬ 
 lations of Philosophy * 
 
 The learned Bede, who lived at the beginning of 
 the 8th century, was acquainted with all the branches 
 of the Mathematics. Under him Alcuin studied, who 
 was well versed in mathematics, and was preceptor to 
 the celebrated Charlemagne. He was a native of Bri¬ 
 tain, where, about this time, more attention was given 
 to the study of this science than in any country in 
 Europe. 
 
 For nearly two centuries after this period, the 
 study of the science was extinct. No mathematician 
 
MATHEMATICS. 
 
 31 
 
 of any eminence appeared till the beginning of the 15th 
 century, when, with the general revival of learning, 
 this science was destined to commence a new and 
 more splendid career. 
 
 The Italians, by their intercourse with Arabia, be¬ 
 came, about the end of the 12th century, the restorers 
 to Christendom of its long-forgotten arts and sciences. 
 Leonardus Pisanus, a rich merchant of Pisa, who had 
 made, in the course of his profession, several voyages 
 to the East, is said to have been the first who dis¬ 
 seminated amongst Europeans a knowledge and a love 
 of mathematical studies, after both had disappeared 
 for so many ages.* 
 
 About the beginning of the 13th century, Campanus, 
 of Navarre, translated Euclid, and several geometri¬ 
 cal works were translated from Greek and Arabic 
 MSS., by Maurolycus, of Messina, who Avas dis¬ 
 tinguished for some original and elegant works of his 
 own. Ramus, or La Raraee, an original metaphysical 
 writer, also composed several excellent works on 
 Geometry : and Willibrad Snellius published at the 
 age of seventeen an attempted restoration of an ancient 
 work of much value. 
 
 About the beginning of the lGth century the 
 Spanish and Portuguese geometers, John of lloyas 
 and Nonnius or Numer, lived. The former Avas the 
 inventor of the projection of the sphere, and the latter 
 Avas the first to introduce the Arabic System of Algebra 
 into Europe. About the same time, Wright, a skilful 
 geometrician, invented his chart—improperly called 
 Mercator’s chart. At this time, Germany produced 
 the three mathematicians, Werner, Rheticus, and Pitis- 
 
 * A MS., dated 1202, of this celebrated Italian, has lately 
 been discovered by Cossali, a canon of Parma. 
 
32 
 
 INTRODUCTION TO THE 
 
 CU8> W erner ma( ^ e a German translation of Euclid. 
 
 I he Italian geometer, Lucus Valerius, determined the 
 centre of gravity in various mathematical figures, 
 conoids and spheroids. 
 
 Kepler was the first to conceive the theory of mag¬ 
 nitude being composed of indefinitely small elements 
 of the same kind; as, for instance, that a circle is 
 composed of the surfaces of an indefinite number of 
 very small triangles having the centre for their common 
 vertex, and their small bases on the circumference. 
 By means of this theory, though not very satisfactory 
 in regard to scientific rigour, he extended Geometry 
 considerably beyond the stage in which it was left by 
 Archimedes. Cavallerius improved on the theory of 
 Kepler, and effected a still further extension of Geo¬ 
 metry by conceiving lines to be composed of points, 
 surfaces of lines, and solids of surfaces. 
 
 Galileo, so eminent for his discoveries in physical 
 science, did little to improve Geometry. He invented 
 the Cycloid, called the Helm of Geometry on account 
 of the discussions that originated concerning its pro¬ 
 perties. Gregory St. Vincent and Andrew Tacquet 
 were two Flemish mathematicians, the former prose¬ 
 cuted the quadrature of the circle—the latter, the 
 quadrature of surfaces and conic sections, with con¬ 
 siderable success. He rendered essential service to 
 the study of the science by subjoining the Uses to the 
 propositions in an edition of Euclid’s Elements, he 
 published, after the manner of the celebrated French 
 Jesuit, Claude Francois Milliett Dechales. Blaise 
 Pascal, a Frenchman, (born 1623, died 1062) 
 was equally distinguished for his great literary and 
 scientific attainments, was the author of several valuable 
 researches respecting the properties of the Cycloid; 
 and at the age of sixteen, he composed a complete and 
 elegant treatise of conic sections. But his health was 
 
MATHEMATICS. 
 
 33 
 
 so much impaired, that soon after he was compelled to 
 suspend his studies for four years ; and he ultimately 
 perished at the age of 39, after a lingering illness, in 
 
 Paris. 
 
 Descartes, another eminent Frenchman, (born 1590, 
 died 1650), had a profound knowledge both of mathe¬ 
 matical and metaphysical science, was one of the 
 most efficient agents in accomplishing that intellec¬ 
 tual revolution which was in progress during his time, 
 and was the inventor of a method which produced a 
 thorough and memorable improvement in Geometry. 
 Before his time the application of Algebra to Geo¬ 
 metry, in the ordinary sense of the term, was known, 
 as examples of it are to be found in the works of some 
 ancient geometers. But Descartes invented that more 
 general mode of applying it—that the science of Geo¬ 
 metry, instead of being confined within the narrow 
 limits imposed by the complexity and inadequacy of 
 the ancient methods, is now of indefinite extent. 
 
 Huygens, a native of Holland, (1629—1705) was 
 a profound mathematician. He solved a problem of 
 such difficulty, that it had not been previously attempted. 
 
 The year 1642, which ‘gathered’ Galileo ‘ to his 
 fathers',’ gave to the world Isaac Newton, the Archi¬ 
 medes of modern times, or, as he has sometimes been 
 styled, from the number and importance of his dis¬ 
 coveries, the Prince of Modern Mathematicians. The 
 fame of Sir Isaac rests chiefly upon the application he 
 made of his acquisitions, in explaining the laws 
 or order by which the mechanism of the universe 
 appeared to him to be regulated. His views with 
 regard to this matter have been styled, in honour of 
 him, the Newtonian System of Philosophy, and have 
 been received implicitly by the learned world tiil nearly 
 the present time. 
 
 The impetus given to mathematical pursuits by 
 
34 
 
 INTRODUCTION TO THE 
 
 the genius of Newton, has descended to our day, 
 to use a phrase of his own, 1 with increasing mo¬ 
 mentum.’ For in his train have followed the illus¬ 
 trious names of Barrow, Keel, Gregory, Bishop 
 Horsley (editor of an edition of Euclid’s Elements), 
 the Rev. Mr. Law’son, Robison, and many others. 
 Among the most celebrated, rank Leibnitz, Euler,* * 
 the Bcrnouillis, Clairant, D’Alembert, f Thomas Simp¬ 
 son, Robert Simson, the well known restorer, and 
 Playfair, the improver, of Euclid—Young, Leslie, and 
 Madame Agnesi,]: and Baron Swedenborg.^ 
 
 It is not necessary to extend this enumeration. The 
 mathematicians and geometers of this generation are 
 known to the public. But it is above estimate how 
 the diffusion of knowledge, one moment dreaded as 
 mischievous, and in the next with the customary in¬ 
 consistency of folly, pronounced impracticable, will 
 increase the public taste for this science, and create 
 a just and enduring appreciation of this useful order 
 of scholars. 
 
 • Euler lost his sight by intense application, and like 
 Milton, closed his labours in blindness. 
 
 + One of the writers in the celebrated French ‘ Encyclo- 
 pajdia.’ 
 
 * l This lady deserves the most respectful mention. Her 
 Algebraical Works, in quarto, translated by Colson, have 
 been long before the public, and very deservedly esteemed.’— 
 Harris's ‘ Compendium of Algebra.’ 
 
 § Swedenborg’s ‘ Principia,’ a work declared, only very 
 lately by one of the highest professors in Europe, * not un¬ 
 worthy of being placed by the side of Newton’s.’— Douglas 
 JerroUl's Mag., vol. 2, page 91. 
 
MATHEMATICS 
 
 35 
 
 CHAPTER IV. 
 
 UTILITY OF MATHEMATICS AS A MEANS OP EXTENDING 
 OUB KNOWLEDGE OP THE PHYSICAL WORLD. 
 
 Plato denied the world can be 
 Governed without Geometry. 
 
 Butler’s Hudibras. 
 
 It is a very trite observation, that human knowledge 
 has been greatly extended by means of this science. 
 Besides the important properties of magnitudes, and 
 wonderful relations of abstract quantities which it has 
 made known, it has unfolded a very extensive ran^e 
 of natural phenomena. It lias investigated the prin¬ 
 ciples of theoretical mechanics, the laws of equilibrium 
 and motion of fluids, fixed and elastic. Optics, elec¬ 
 tricity, and magnetism are its debtors. The theory of 
 acoustics, the propagation of light, and numerous 
 other branches of science, have received great im¬ 
 provements by its agency. Even its reputed conceits 
 have been susceptible of useful application. The 
 ancient doctrine of the Conic Sections, which for 2000 
 years was an object of mere curious speculation, be¬ 
 came, in the hands of Newton, a most efficient means 
 of unfolding the planetary motions. 
 
 ‘ Without the aid of rules derived from this science, 
 the navigator, relying on his compass as a guide, could 
 not with safety venture to any considerable distance 
 on his element; intercourse with transmarine regions 
 would be impossible; and, consequently, our know¬ 
 ledge of the globe we inhabit would be very limited. 
 We should probably yet believe that its surface is an 
 extended plane, and that it is supported on pillars; or, 
 as was the opinion of some ancient philosophers, that 
 
36 
 
 INTRODUCTION TO THE 
 
 its figure is cylindrical, like a drum. Without the aid 
 of this science, our knowledge of celestial bodies 
 would be still more imperfect, and the consequences of 
 our ignorance still more striking. We should still 
 believe that these objects are equally distant from us, 
 and, very probably, that they are distributed on the 
 surface of an extensive crystalline sphere, performing 
 a diurnal rotation about the earth, as the centre of the 
 universe. We should also believe that some celestial 
 phenomena, as eclipses and comets, are certain signs 
 of a conflict of the elements of nature, or that they 
 are the portentous indications of the wrath of heaven, 
 while contemplating to inflict on superstitious mortals 
 some dire calamity, as war, pestilence, or famine. 
 
 ‘ How different from these unsatisfactory and inco¬ 
 herent conjectures is that great achievement of this 
 science—the clear and satisfactory exposition, on the 
 most incontrovertible principles, of the complex though 
 sublime and systematic mechanism of the heavens; 
 by which the distances and magnitude of the sun and 
 planets have been measured, and also their weights, 
 and even that of their satellites, ascertained; and by 
 which the masses and distances of some of the stars, 
 or suns of other systems, though inconceivably remote, 
 even in comparison with the great extent of our own 
 system, will probably ere long be determined/* 
 
 An abler pen has celebrated this theme. ‘ The 
 scientific principles that men employ to obtain the 
 foreknowledge of an eclipse, or of anything else 
 relating to the motion of the heavenly bodies, are 
 contained chiefly in that part of science which is called 
 Trigonometry, or the properties of a triangle, which, 
 when applied to the study of heavenly bodies, is called 
 Astronomy; when applied to direct the course of a 
 
 * Bell. 
 
MATHEMATICS. 
 
 37 
 
 ship on the ocean, it is called Navigation; when ap¬ 
 plied to the construction of figures drawn by rule and 
 compass, it is called Geometry; when applied to the 
 construction of plans of edifices, it is called Archi¬ 
 tecture ; when applied to the measurement of the 
 surface of any portion of the earth, it is called Land- 
 surveying. In fine, it is the soul of science. It is an 
 eternal truth. It contains the mathematical demon¬ 
 stration of which man speaks, and the extent of its 
 uses is unknown.’* 
 
 The author of the Geometrical Companion defends 
 the practical uses of Geometry with the ardour of an 
 enthusiast. His examples are happy and forcible. 
 
 * A bee, which is an instinctive geometer, finds the 
 use of it in constructing its comb, so as to be the most 
 roomy and durable at the least expense of time and 
 labour. Will any one after this fact doubt the use of 
 Geometry to an intelligent creature ? 
 
 ‘ Almost all the evolutions of an army consist in 
 converting one parallelogram into another. And the 
 works of a fortified town, or position, are little else 
 than combinations of parallelograms forming bastions, 
 curtains, and parapets.’f 
 
 In our ‘ Uses,’ other applications of this science will 
 be seen. Perspective, Dialling, and a variety of the 
 arts, might further be adduced. Literature gathers 
 from it many of its prettiest similes. Dr. Johnson has 
 two examples in his Life of Addison :— 1 A simile may 
 be compared to two lines converging to a j)oint, and 
 it is more excellent as they approach from a greater 
 distance.’ 
 
 ‘ An exemplification resembles two parallel lines, 
 which run on together without approximation, never 
 far separated, and yet never joined.’ 
 
 * Paine. + Darley. 
 
33 
 
 INTRODUCTION TO THE 
 
 Emerson has a fine thought adorned with the 
 vestments of geometry. ‘ Every natural process is 
 but a version of a moral sentence. The moral law 
 lies at the centre of nature, and radiates to the cir¬ 
 cumference.’ 
 
 CHAPTER V. 
 
 MATHEMATICS AS A MEANS OF MENTAL DISCIPLINE. 
 
 The same excellent persons in the commonwealth of 
 learning who discovered Philosophy discovered also the Ma¬ 
 thematics : like twin sisters they came into the world, and 
 like twin sisters there is a natural affinity between them.— 
 Tacquet. 
 
 ^t is a mortifying reflection that the soberest of 
 sciences should have been disfigured by the exaggera¬ 
 tions of untutored enthusiasm. The Mathematics 
 have been extolled as the only source of logical 
 acumen and cool judgment. Its discipline has been 
 assumed to be the most salutary to which the human 
 powers can be subjected. Some indiscreet votaries 
 have carried its principles out of their legitimate pro¬ 
 vince, and applied them to subjects they were never 
 fitted to investigate. As might be expected, these pro¬ 
 ceedings have led to the disappointment of those who 
 believed such unwarranted pretensions, and to the ulti¬ 
 mate undervaluing of the real and substantial merits of 
 Mathematics. No more than the varying constitutions 
 of mankind admit of one panacea lor human ailments, 
 do the variations in human intelligence admit the pre¬ 
 tensions of one science to train up the faculties of every 
 individual. Natural Philosophy, Political Economy, 
 Morals, Metaphysics, each has its peculiar discipline 
 by which it addresses peculiar persons, and matures 
 
MATH KM AT ICS. 
 
 39 
 
 their powers where Mathematics would entirely fail. 
 The discreet friends of Mathematics never hesitate to 
 allow this, and with the judicious discrimination of 
 Tacquet, care only to claim for it kindred affinity to 
 Philosophy, and the moderate hut just honour of its 
 having shed a certain lustre over human manners, and 
 the arts and the sciences at largo. 
 
 Dr. Amott, author of the Elements of Physics, 
 has attacked the assumed pretensions of Mathema¬ 
 tics, and though he is far from doing it skilfully, it 
 cannot be denied that he had occasion for his reflec¬ 
 tions. * There is/ he says, 1 no occupation which so 
 much strengthens and quickens the judgment as the 
 study of Natural Philosophy. This praise has usually 
 been bestowed on Mathematics; yet a knowledge of 
 abstract Mathematics existed with all the absurdities 
 of the dark ayes; but a familiarity with Natural Phi¬ 
 losophy, which comprehends Mathematics, and gives 
 tangible and pleasing illustrations of its abstract 
 truths, seems incompatible with any gross absurdity. 
 A man whose mental faculties have been sharpened by 
 acquaintance with these exact sciences, in their com¬ 
 bination, and who has been engaged, therefore, in 
 contemplating real relations , is more likely to discover 
 truth in other questions, and can better defend himself 
 against sophistry of every kind. We cannot have 
 clearer evidence of this, than in the history of the 
 sciences, since the Baconian method of reasoning by 
 induction took the place of the visionary hypotheses of 
 preceding times.’ 
 
 Dr. Arnott does not seem an enemy to pretension, pro¬ 
 vided he is allowed to advance it. The kind of emi¬ 
 nence he denies to Mathematics, and which might safely 
 be denied of any science, he does not hesitate to claim 
 for Natural Philosophy. He unhesitatingly asserts 
 that ‘ there is no occupation which so much strengthens 
 l) 
 
40 
 
 INTRODUCTION TO THE 
 
 the judgment as the study of Natural Philosophy.' 
 But common experience, in the variety of intellectual 
 aptitudes, teaches us the fallacy of this broad asser¬ 
 tion. It is not a disparagement, as Dr. Arnott sup¬ 
 poses, that 1 Mathematics existed with all the absurdi¬ 
 ties of the dark ages;’ on the contrary, it is no little 
 praise that Mathematics had vitality enough to exist at 
 all under that weight of superstition and barbarism, 
 which buried all other learning in undistinguished 
 ruin. 
 
 A geometrician, whose ingenious performances I 
 shall have several occasions to cite, and whose popular 
 pen will give him considerable influence in other 
 quarters, has contributed a eulogium to his favourite 
 science, which cannot be permitted to pass unnoticed. 
 
 1 Geometry not only sharpens our faculties, corrects 
 precipitancy, and prevents liability to imposition, either 
 from the arguments of others or our own, but it gives 
 us a satisfaction in our knowledge [which we could 
 not otherwise obtain], and thereby enables us to speak 
 with a strength and perspicuity which a vague per¬ 
 suasion of the truth of what we utter could never 
 inspire.’ 
 
 Mr. Darley affirms too much. He plainly alludes to 
 ‘ satisfaction’ in knowledge in general, and the words I 
 have put in brackets should have been omitted, as men 
 can, for thousands do, ‘ otherwise’ succeed in securing 
 that satisfaction.* It is enough to contend, which 
 
 * It is but fair to this writer to state that he has deduced 
 a compliment to Geometry from the biography of distin¬ 
 guished men, expressed with perfect propriety:— 
 
 ‘ Alexander, Caesar, Charles 12th, Buonaparte, and two of 
 the most celebrated military engineers, Yauban and Coehorn, 
 were profound geometers. And it is to their knowledge of 
 this science, the superior excellence of their systems must be 
 attributed .’—Geometrical Companion, 
 
MATHEMATICS. 
 
 41 
 
 -would l>e true, that the study of Geometry is one of 
 the inodes of educational discipline, and a most ap¬ 
 proved method indeed. It is contending that it is the 
 only method which discovers vanity and offends good 
 
 taste. 
 
 When a stranger, an enemy, or an indifferent person, 
 chooses to laud Mathematics to the disparagement of 
 other branches of knowledge, the encomium may he 
 tolerated, though by no means imitated. Dugald 
 Stewart, who had no particular predilection for the 
 Mathematics, thus speaks of its study in his Philoso¬ 
 phy of the Human Mind :— 
 
 ‘ The intellectual habits of the metaphysician afford 
 little or no exercise to that species of attention which 
 enables us to follow long processes of reasoning, and 
 to keep in view all the various steps of an investiga¬ 
 tion till we arrive at the conclusion. Such processes 
 are much longer in Mathematics than in any other 
 science; and hence the study of it is peculiarly calcu¬ 
 lated to strengthen the power of steady and concatenated 
 thinking ; a power which, in all the pursuits of life, 
 whether speculative or active, is one of the most valu¬ 
 able endowments we can possess.’ 
 
 The Geologist, the Astronomer, the Chemist, seve¬ 
 rally have lauded their respective sciences as the most 
 important that can engage the attention of mankind, 
 both in an absolute and educational sense. This is the 
 language of ignorance or vanity, Mathematics need 
 never descend to the disparagement of other sciences. 
 It has its distinctive merits: it can bear any com¬ 
 parison, its province is peculiar, its utility unques¬ 
 tioned, and its method of procedure instructive. 
 These facts are its proper eulogies, and time has 
 failed to obscure, or the envy of ages to diminish, 
 their lustre. 
 
 Some of the features of discipline which indisput- 
 
42 
 
 INTRODUCTION TO THE 
 
 ably characterise mathematical study, have been well 
 depicted by a modern writer. 
 
 ‘ In reasoning, as in other arts, we are not masters of 
 what we do, till we do it both well and unconsciously: 
 now this advantage a judicious cultivation of Ma¬ 
 thematics supplies. It familiarises the student with 
 the usual forms of inference, till they find a ready 
 passage through the mind, while anything which is 
 fallacious and logically wrong, at once shocks his 
 habits of thought, and is rejected. He is accustomed 
 to a chain of deduction, where each link hangs on the 
 preceding; and thus he learns continuity of attention 
 and coherency of thought. His notice is steadily 
 fixed on those circumstances only in the subject on 
 which the demonstrativeness depends, and thus that 
 mixture of various grounds of conviction which is so 
 common to other men’s minds is rigorously excluded 
 from his. He knows that all depends upon his first 
 principles, and flows inevitably from them; and that 
 however far he may have travelled, he can at will 
 go over any portion of his path, and satisfy himself 
 that it is legitimate; and thus he acquires a just per¬ 
 suasion of the importance of principles on the one 
 hand, and, on the other, of the necessary and constant 
 identity of the conclusions legitimately deduced from 
 them.’* 
 
 Dr. Olinthus Gregory, in his farewell address to the 
 pupils of the Royal Military Academy, speaks of the 
 study of Geometiy as leading to that valued attain¬ 
 ment—‘ The power of mastering the mind. If it be 
 desirable to obtain and keep the ascendancy anywhere, 
 it is, surely, at home , in the centre of your own intel¬ 
 lect and its principles, your heart and its emotions. 
 
 * Thoughts on Mathematics as a part of Liberal Educa¬ 
 tion, by the Rev. William Whewell, Cambridge. 
 
MATHEMATICS. 
 
 43 
 
 Gain the mastery here, govern within, learn to direct 
 your thoughts to any subject you please, and keep 
 them uninterruptedly to their occupation, till at your 
 bidding the labour shall be remitted : then all will be 
 well, for all will lead to a prosperous issue.’ 
 
 We all know, that if human legs made peregrina¬ 
 tions without the consent of the owner, how great would 
 be the hindrance to business, yet how often is thought 
 found roaming the world over when it should be con¬ 
 fined at home But the command of the thoughts is 
 as essential to success in thinking as the control of the 
 limbs is to the dispatch of business. Inasmuch as 
 Mathematics imparts this power, it is valuable dis¬ 
 cipline. 
 
 In the same address Dr. Gregory justly remarks, 
 that the power of doing one thing at one time is the 
 hardest and the most important lesson a young man 
 has to learn. He is of opinion that the concentration 
 of attention and good sense included in this habit, is 
 powerfully impressed upon the notice of the mathe¬ 
 matical student. The wisdom of this course, as ap¬ 
 plicable to children as well as adults, has lately been 
 forcibly illustrated by a lady in these words :—‘ A 
 very quick and clever child made an observation to her 
 governess before me the other day, which had 
 much of truth in it. “ How is it, my dear,” inquired 
 the lady, “that you do not understand this simple 
 thing?” “ I do not know, indeed,” she answered, 
 with a perplexed look, “but I sometimes think I have 
 so manv things to learn that I have not time to under¬ 
 stand.”'’ 
 
 If we may trust the testimony of another enthu¬ 
 siast, this science exercises a species of influence 
 of a very high order. ‘ The moral influence of the 
 Mathematics did not escape the penetration of Plato, 
 who, chiefly on that account, decides that the study 
 
u 
 
 INTRODUCTION TO THE 
 
 should be cultivated in his “Republic.” “ Geometry,” 
 says he, “ is the knowledge of that which is eternal; 
 it disposes the mind to the contemplation of truth, and 
 gives a philosophical habit, which withdraws our 
 thoughts from what is low to what is elevated. Let 
 not the citizens of this renowned Republic neglect the 
 study of Geometry.”’ Mr. Cooley concludes w r ith 
 observing — 1 Wc w r ell know how much it conduces to 
 the easy acquisition of any kind of knowledge, to have 
 previously learned Geometry.’* 
 
 But it is far from my intention to rest the repu¬ 
 tation of geometrical studies upon declamation, how¬ 
 ever eloquent, or upon testimony however respectable. 
 An exposition of the Logic of Euclid is the best demon¬ 
 stration of the value of its discipline, and this I shall 
 now attempt. 
 
 CHAPTER VI. 
 
 TIIE LOGIC OF EUCLID. 
 
 * Among the various guides which have been written for 
 the direction of the understanding, no system of logic has 
 ever been devised better calculated for this purpose than the 
 Mathematics .’—James Harris: ‘Introduction to Compendium 
 of Algebra.’ 
 
 Moderating somewhat the absolute tone of Mr. 
 Harris’s opinion, it may be taken as expressive of the 
 truth. The ancients considered Euclid their best book 
 of logic. Its main features as such are, that from a few 
 axioms, self-evident, and definitions, agreed upon, new 
 truths are evolved by the perception of the coherence 
 
 * Cooley’s Introduction to the Study of Mathematics. 
 
MATHEMATICS. 
 
 45 
 
 subsisting between them and the things previously 
 admitted. These truths are added in their turn to the 
 ranks of axioms from which others are to l e interred. 
 Until this day sciences are considered perfect in pro¬ 
 portion as they possess these features.* To Geometry 
 belong three principal things:—Axioms, or first prin¬ 
 ciples—Definitions, or explanations of the meanings of 
 the chief terms employed—Postulates, or things re¬ 
 quired to be granted. 
 
 An enquirer must understand himself, and disputants 
 must understand each other, or efforts after convic¬ 
 tion will be futile. The geometrician (for we shall use 
 him as the exponent of mathematical procedure) 
 commences by distinguishing what he must assume : 
 for such is the limited nature of human intellect, that 
 all reason must primarily rest on assumption. Ac¬ 
 cordingly, Euclid lays down axioms which, being, as 
 their name implies, self-evident, become the com¬ 
 mon point from which disputants can start on the path 
 of investigation. In geometrical treatises, Definitions 
 are usually placed first, but in logical propriety 
 Axioms demand first to be considered, as llie assent to 
 definitions is often made by a secret reference to 
 axioms in the thoughts. 
 
 AXIOMS. 
 
 The first principles, or fundamental axioms com¬ 
 monly used in Geometry are these :—‘ If a thing be 
 divided into parts, the whole is greater than any one 
 of the parts. Two straight lines cannot enclose a 
 space. All lines drawn from the centre to the circum- 
 
 * Mr. James Milne, in a curious work on National Educa¬ 
 tion, has constructed outlines of Political Economy, Meta¬ 
 physics, and other subjects, on these principles. Though 
 not entirely successful, they are instructive speculations. 
 
40 
 
 INTRODUCTION TO THE 
 
 fcrence of ft circle, are equal to each other. Two things 
 cannot be both equal and unequal at the same time. 
 Things equal to the same thing are equal to one an¬ 
 other.’* The information conveyed by such elemen¬ 
 tary truisms is necessary : they are the very A, B, C, 
 of the science—the lowest steps of the mathematical 
 ladder, and as essentially useful as the highest. 
 
 The patient reference which the geometer makes to 
 these first principles is highly instructive. He never 
 loses si'dit of them. At his highest altitude he turns 
 to refresh his understanding at those clear fountains of 
 sense—an example which every rcasoner may imitate, 
 and an advantage which the investigator in every 
 science may profitably secure to himself. 
 
 Dr. South has encouragingly remarked, that the 
 reason of all things lies in a narrow compass.f The 
 geometrician is professionally master of this secret, so 
 true of every rational theme of human speculation. 
 An Iliad of meaning lies concealed in the nut-shell of 
 a fundamental axiom. Of what importance is this 
 truth to the enquiring public of our day ? 
 
 ‘ How manv readers are there who would not be 
 glad of attaining to knowledge the shortest way, see¬ 
 ing the orb thereof is swollen to such a magnitude, 
 
 * It strikingly evidences the intimate connection between 
 Logic and Geometry that this axiom and its converse are 
 found at the root of Aristotle’s Syllogisms. Syllogisms 
 were deducible from these two principles—First: things 
 which agree with the same thing, agree with one another. 
 Second: things whereof one does, and one does not agree 
 with the same thing, do not agree with one another.— Bruce. 
 
 + The reason of things lies in a narrow compass. Most of 
 the writings in the world are but illustration and rhetoric, of 
 little consequence to a mind in pursuit after the philosophical 
 truth.— Dr. South. 
 
MATHEMATICS 
 
 4*5 
 
 and life but such a span to grasp it? How many who 
 have not some curiosity to know the foundations of 
 those tenets upon which they so securely trust their 
 understanding ? or where the footsteps of those opin¬ 
 ions and precedents may be found, which have given 
 direction to so many modern performances ? * 
 
 No science is fully mastered till the principles on 
 which it is based are clearly seen. When these are 
 lost sight of, all subsequent reasoning is confused. The 
 geometrical student is early and much accustomed to 
 keep these important considerations before him. The 
 most abstruse proposition is an axiom to the mind 
 that perceives its truth, and the most complicated 
 science resolves itself into a very simple affair when 
 its first principles become familiar to the understand¬ 
 ing. These considerations reduce learning to an art, 
 and give tone, power, and comprehensiveness to the 
 understanding, which fit it peculiarly for the acquisi¬ 
 tion of knowledge. 
 
 Ordinary experience in learning and teaching must 
 frequently impress the truth of these remarks on care¬ 
 ful thinkers. When a pupil has a voluminous gram¬ 
 mar put into his hands, lie feels, with Dr. Johnson, that 
 a great book is a great evil. But when the first 
 principles of language are clearly seen, the difficulties 
 vanish and the study is mastered. 
 
 This is not only the secret of the acquisition of know¬ 
 ledge, but also the agent for retaining it. Around these 
 first principles, as around a standard, the thoughts natu¬ 
 rally associate. Touch but a remote chord of any ques¬ 
 tion, and it will vibrate to the central principles to which 
 it hasonce been well attached—every relative impression 
 owns a kindred connection, and the moment one is 
 attacked, it, like a faithful sentinel, arouses a whole 
 
 * Oldys. 
 
48 
 
 INTRODUCTION TO THE 
 
 troop, who, marshalled and disciplined, bear down 
 with their whole strength, and challenge the enemy. 
 
 DEFINITIONS. 
 
 That confusion in the varying meaning of terms, 
 which reigns through most subjects that agitate 
 mankind, is unknown in Geometry. The Definitions 
 at the very threshold for ever fix the question of sense. 
 Every leading term is defined. There are no words 
 used in Geometry whose meanings are so much 
 alike, that the ideas for which they stand may be con¬ 
 founded together. If there is any distinction between 
 the meanings of two terms, it is a total and a complete 
 distinction. Locke, in illustration of this point, 
 remarks —‘ The idea of two, is as distinct from the 
 idea of three, as the magnitude of the whole earth is 
 from that of a mite ; this is not so in other modes, in 
 which it is not so easy, nor perhaps possible for us to 
 distinguish between two approaching ideas, which yet 
 are really different; for who will undertake to find a 
 difference between the white of this paper, and that of 
 the next degree to it ?’ 
 
 The meanings once fixed in Geometry are ever 
 after adhered to : the same word is for ever used in 
 the same sense. For instance, the term right signifies 
 straight, and straight is its everlasting meaning in 
 Geometry. In political philosophy, right is a well- 
 known stumbling block, which perpetually overthrew 
 the fine genius of Burke. ‘ Natural right/ ‘ inherent 
 right/ ‘ moral right/ and ‘ political right,’ are only a 
 few of the rights which every now and then some 
 puzzled politician will pronounce wrong. It perhaps 
 is not possible to restrict popular phrases as we can 
 scientific terms. But the geometrician, perceiving 
 the advantages of doing so, acquires the taste of 
 precision, and takes care (which is practicable with 
 
MATHEMATICS. 49 
 
 everybody) to define the principal terms he may em¬ 
 ploy in any given dissertation. 
 
 POSTULATES. 
 
 With that precision from which the geometer never 
 departs, he defines his mode of procedure. For this * 
 purpose he puts up his Postulates or petitions to be 
 permitted a certain latitude of action. This latitude 
 is never exceeded. Thus neither his language nor 
 his practice is left open to cavil. Admit my axioms, 
 agree to my definitions, and grant my Postulates, 
 says the independent geometer, and I will engage to 
 demonstrate every proposition I lay down. 
 
 A proposition to be proved, or a problem to be 
 solved, have common parts—the Proposition, Enun¬ 
 ciation, Construction, Demonstration, Conclusion. 
 (These parts, for the assistance of the uninitiated, are 
 clearly distinguished in the following pages.) * 
 
 The proposition being plainly laid down, the enun¬ 
 ciation fixes its meaning, the construction provides for 
 its proof, and a rigid demonstration leads the way to 
 the conclusion. Every step has its distinct reference, 
 establishing its legality and fairness. There is no 
 attempt at evasion—no manoeuvre, no equivocation. 
 This daring self-reliance cannot fail to impart manly 
 qualities. 
 
 The student is distracted with no impatient attempts 
 to accomplish two things at once. One thing at one 
 time is the order of Geometry. In the language of the 
 propositions, all circumlocution is avoided. What is 
 intended is straightway expressed. No digressions 
 
 * One who does not understand the principles of Euclid’s 
 Demonstrations, knows absolutely nothing of Geometry: unless 
 he attain this point, all his labour is utterly lost.— Dr. Wkate.y. 
 Preface to Logic,’ p. xxi. 
 
60 
 
 INTRODUCTION TO THE 
 
 are endured. Nothing is admitted but that which has 
 been sanctioned, or established, or is relevant. Nothin? 
 is asserted but that which is intended to be demom 
 strated. It is impossible that these methods of pro¬ 
 cedure can be perceived and practised by the student 
 'without his acquiring valuable lessons, applicable to 
 other subjects, and to much of the conduct of life. 
 
 The reader, in this place, must be cautioned against 
 the language in which the Logic of Euclid is some¬ 
 times eulogised. Here is an extract from a recent 
 journal:—‘ Never does the mathematician say, such a 
 proposition is true because it is probable; or, because 
 the balance of the pro and con arguments preponderates 
 in its favour ; or, because history records it, and tradi¬ 
 tion affirms it; or, because the holy St. This declared 
 it, and the learned Philosopher That believed it; or, 
 because infallible I assert it. No ! hr takes a nobler 
 stand, he claims more lofty ground ; his words are — “ I 
 have actually and incontrovertibly proved so and so to 
 be a fact; disbelieve me, if you can; refute me, if you 
 arc able.” ’ This is the folly that encourages the con¬ 
 tempt with which this fine study is sometimes treated.* 
 The reason why the geometer neither refers to 
 probability nor authority is, that the nature of his 
 science enables him to do without them. In the 
 affairs of mankind there are numerous vital questions, 
 such as those of law and legislation, which, from 
 the nature of things, must be decided by probability: 
 by pro and con arguments, and such decisions are as 
 respectable and as salutary, if not quite as safe, as 
 the conclusions of Mathematics. Those speculations 
 
 .1 f It was this thoughtless arrogance which provoked the 
 
 HUM Oicfupt from a recent critic—■* This logical and 
 
 mathematical school of literature can only hold its sway with 
 narrow minds and in an unimpassionedage .’—Douglas J err old's 
 ‘ Shilling Magazine,’ vol. 1, p. 572. 
 
MATIIEMA1 If S. 
 
 51 
 
 •—and others, curious and important ones, in which 
 authority and personal conviction for a long time 
 form the only accessible evidence—are not to be re¬ 
 jected on that account, but to such evidence is to be 
 allowed its proper weight. Let not flippant arrogance, 
 rejoicing in the strength of Geometry, depreciate the 
 arguments peculiar and necessary to other subjects. The 
 modest mathematician learns the lessons of valuing 
 strict demonstration wherever it can be had, and rightly 
 refuses conjecture where certainty is to be obtained. 
 He knows, if lie has general information, that every 
 proposition cannot be proved after the manner oi 
 Geometry, but he has a right to expect that what 
 is attempted to be demonstrated in the same way shall 
 be demonstrated with equal strictness. 
 
 Geometiy is theoretical and practical : demon¬ 
 stration, direct and indirect: and reasoning is syn¬ 
 thetical and analytical. Grammar is an illustration 
 of the synthetical ; Chemistry of the analytical. In 
 Grammar we begin with letters—these are combined 
 into syllables — syllables into words — words into 
 sentences—and sentences into discourses. This is 
 synthesis, or putting together. In Geometry, the 
 grand element of the whole superstructure is a point, 
 a point becomes a line, a line a surface: and of these 
 solids are composed. Geometry is synthetical. In 
 Chemistry, water is decomposed and resolved into 
 its elementary principles by means of analysis, or 
 taking to pieces. This is a similar demonstration, 
 though obtained by a method the reverse of the former. 
 There is a third method—namely, Induction, which 
 Hume, with more precision than Locke, calls tho 
 doctrine of proofs. Induction is inferring from several 
 particular facts, or propositions, a general truth. All 
 these modes have important uses in different investi¬ 
 gations. Geometrical reasoning is inductive. The 
 
•52 
 
 INTRODUCTION TO THE 
 
 Axioms and properties of Definitions are the facts from 
 which first conclusions are deduced. Every pro¬ 
 position established, is an addition to these facts, which 
 become the increased bases of inferences. From 
 coherency between ideas, the geometrical logician pro¬ 
 ceeds to coherency between facts, and from the entire 
 body of facts before him he discovers the coherency of 
 his general inference. Reason is the ability to 
 perceive coherences. Reasoning on the abstrusest 
 theorems is nothing more than after having arrived, 
 step by step, at a remote truth, discovering its connec¬ 
 tion Avith preceding facts in the same chain. Geo¬ 
 metrical problems alarm the uninitiated, who are not 
 aware that a little attention conquers the simple ones, 
 and a little patience the most difficult. One in the 
 secret thinks them all equally easy, because he sees 
 they are all equally connected. So far as geometrical 
 study affords practice in mastering these consecutive 
 coherences, it may be considered as aiding the de¬ 
 velopment of reasoning power. 
 
 The ‘ First Book of Euclid ’ is a perfect illustration 
 of geometrical logic. It embraces the foundation of 
 all the subsequent books. They deal with new pro¬ 
 positions and advanced reasoning, but the principle of 
 procedure is the same as in the ‘ First Book.’ So 
 when the inquirer understands that, he comprehends 
 the principle of them all. This ‘First Book ’ furnishes 
 general principles of reasoning, which will carry the 
 student, without other assistance, through every subject 
 to which his attention can be called. He has only to 
 use them as initiative guides—giving him a taste for 
 strictness in every application—not as fettering him to 
 one species of evidence and inference. The discretion 
 here alluded to is that so well enforced by the author 
 of Hermes. ‘ When Mathematics, instead of being 
 applied to cxemjilify Logic, comes to supply its place, 
 
MATHEMATICS. 
 
 no wonder if it pass into contempt. For when men, 
 knowing nothing of that reasoning which is universal, 
 come to attach themselves for years to a single sjxrics, 
 a species involved in lines and numbers only, they 
 grow insensibly to believe these last ns inseparable 
 from all reasoning, as the poor Indians thought every 
 horseman to be inseparable from his horse/ 
 
 CHAPTER VII. 
 
 NATURAL GEOMETRY. 
 
 Mr. Macaulay, in his historical Essay on Lord 
 Bacon, has some ingenious remarks on the natural 
 origin of the species of Induction with which Bacon's 
 name is associated. Similar opinions have been ex¬ 
 pressed respecting the Syllogism of Aristotle. One 
 of the ablest opponents of John Locke has some 
 profitable speculations of the same kind concerning 
 reasoning in general, all bringing Science, Art, and 
 Logic home to nature, and resting their foundation 
 there. This is the most encouraging aspect of learn¬ 
 ing that can be presented to the notice of a pupil. 
 No treatise of an educational nature should be 
 without such a feature. If an inquirer once per¬ 
 ceives that the germs of a desired art are familiar 
 to him, he makes an important step in advance— 
 he gains confidence. He finds that he has only to 
 systematise and extend his actual knowledge. The 
 barrier of exclusiveness is broken down, and he enters 
 the common ground of the initiated. A geometrician 
 before referred to, has furnished some interesting illus¬ 
 trations of this truth with respect to his lavounte 
 science, accompanied by not less useful remarks. 
 
34 
 
 INTRODUCTION TO THE 
 
 * Suppose Codrus died, and left his field, nearly a square, 
 to be divided among his four sons, Caius, Gaius, Titius, and 
 Tatius. A little consideration would show that it might 
 be done by drawing lines from the middle of one side to the 
 middle of the other, as A B, C D, (see fig. 1). Thus there 
 would be four squares of one size for the brothers.' The 
 middle point would perhaps be found by stepping along one 
 side, and half the number of steps back. Or with a line or 
 a rod. Such inodes would be practical, and such as would 
 be suggested to the simplest mind—for the rudiments of 
 Geometry must be in the human mind before the art itself 
 could be formally known as a system. 
 
 ‘ Suppose the field to be three cornered (see fig. 2). A 
 little more (though not much) sagacity would perceive that 
 its division would be effected by drawing a line from any 
 corner D of the field to the middle point A of the opposite 
 side—and from A to C and B, the middle points of the 
 remaining sides. 
 
 ‘ But if the field had been of the unmanageable form as 
 fig. 3, something more than common judgment would be 
 necessary to subdivide this into four equal portions. Some¬ 
 thing would be wanted beyond what the senses or unculti¬ 
 vated reason could furnish, to recognise the equality of the 
 parts A, B, C, D, when the figure had been accidentally 
 divided as required. Then men would begin to consider 
 how they might determine the equality or non-equality of 
 differently shaped figures. They would, as we are told was 
 the practice of the early philosophers, draw figures in sand 
 or dust to investigate their proportions: first, because it 
 was useful, and afterwards, because curious. This would 
 be land measurement in miniature. The next removal 
 would be from the sand to the slate, from the dust to the 
 board, and from these to parchment or paper. What before 
 engaged peasants without doors, would now employ philo¬ 
 sophers within: what was rude, vague, and unsatisfactory, 
 would now become accurate, definite, and conclusive • what 
 was done experimentally with the rod and the tape, would 
 now be determined rationally with the rule and compass. 
 Behold the gradual transformation of the practical into the 
 abstract art—of primitive into refined Geometry ! 
 
 Ask a ploughman how he manages to keep his furrows 
 
MATHEMATICS. 
 
 ■~o 
 
 parallel, and lie will tell you, by driving his team and 
 directing the share, so that each new furrow shall keep at 
 the same distance from the last, throughout the whole 
 length. In like manner the school boy rules his copy 
 straight by keeping each end of the ruler at the same dis¬ 
 tance from the last line ruled. Thus is the doctrine ■ f 
 parallels found inherent in the commonest minds. 
 
 * Let us by all means get rid of the scholastic notion, that 
 Geometry is a thing totally distinct from, and independent 
 of, the material world—irrelevant to the ordinary business of 
 life, and uncongenial with the common trains of human 
 reflection : that it is, in fact, an ethereal system, not refined 
 from the grossness of terrestrial conceptions, but absolutely 
 free from all intermixture or affinity with what we see or 
 know about earthly matters: and that it is drawn from the 
 skies, as Socrates drew his moral philosophy, and is a study 
 for those alone who agree to call each other “ purely in¬ 
 tellectual beings.” No such thing. It is only the simplest 
 and most certain of our ideas, gathered from practice and 
 experience, rendered accurate by strict circumspection, and 
 combined with elegant ingenuity into a system differing 
 from others of human invention only in being more perfect. 
 So far is it from being anything of a supernatural or occult 
 science, that a man of strong judgment might, in a popular 
 way, form to himself a system of Geometry almost without 
 knowing how to read. In fact, a great many illiterate men 
 do so. What a powerful system of natural Geometry must 
 the celebrated Watt have organised in his mind ! lie was 
 ignorant of this science (that is, of the written word of it), 
 but he was as well acquainted with the common truths of 
 it as the most deeply initiated mathematician, or he would 
 never have made his great discoveries, which involve those 
 properties.’ * 
 
 A little reflection on the Elements of Geometry, as 
 displayed in the following pages, will further show 
 that the principles of the science have their rise in 
 human experience. The * cutting off the corner’ of a 
 street in walking evinces an intuitive knowledge of a 
 
 * Abridged from Barley's Geometrical Companion. 
 
INTRODUCTION TO TIIE 
 
 50 
 
 geometrical theorem. The elements are common to all 
 comprehensions—Science develops and perfects them. 
 
 CHAPTER VIII. 
 
 nt\CTICE AND THEORY—OR, THE DISTINCTION BETWEEN 
 PRACTICAL GEOMETRY AND TUBE MATHEMATICS. 
 
 Pure Geometry is theoretical. Euclid’s Elements 
 are considered pure Mathematics. In this species of 
 Geometry everything is done in accordance with the 
 Postulates of Euclid. No instruments, nor compass, 
 nor rule, are used, and the results arrived at are all 
 abstract and unapplied. 
 
 Practical Geometry, on the other hand, is seen in 
 the constructions connected with the Propositions, and 
 in the ‘ Usea’ annexed to our ‘ First Book.’ In Prac¬ 
 tical Geometry, compasses, sections, rules—all species 
 of instruments are employed. It is the application of 
 Geometry to the arts of life. 
 
 In pure Geometry something of the practical cer¬ 
 tainly appears in the ‘ Constructions ’ of certain of 
 the Problems and Propositions, but is employed no 
 farther than is necessary for the development of 
 the demonstration—the practical is subordinate, while 
 in Practical Geometry it is the essence. Pure Geo¬ 
 metry chiefly demonstrates that a given Proposition is 
 true, or that a given thing can be done. Practical 
 Geometry, on the other hand, teaches all convenient 
 methods of practice, and applies it to any or every¬ 
 thing to which it can be useful. Such is the aptitude 
 of human intelligence for reality, that when we assume 
 to apply only the Postulates of Euclid, we think of 
 the instruments; but this difference still remains—in 
 Practical Geometry the rule and compass are in the 
 hands , in pure Geometry only in the thoughts. 
 
MATHEMATICS. 
 
 o7 
 
 In this work it is attempted to exhibit the ‘ Beauties’ 
 of Euclid’s Elements theoretically , and the ‘ Uses’ 
 practically —because where the theory only is ex¬ 
 hibited the question is heard—‘ It is all very beau¬ 
 tiful, but what is the use of it ? ’ If only the 
 practice is shown, the remark arises —‘ I see it is so, 
 but I do not see why.’* The question and the remark 
 are both proper, and both should be met. These dis¬ 
 tinctions rule throughout Mathematics. Whatever 
 chiefly pertains to demonstration, and shows that ;t 
 certain thing can be done, or that a certain result will 
 ensue, belongs to the province of pure Mathematics 
 —while in Practical Mathematics attention is mainly 
 devoted to modes of operation and to the application 
 of principles. In the pure department, all is abstraction 
 and theory—in the other, all application and practice. 
 
 CHAPTER IX. 
 
 EXORDIAL ADDRESS TO TI1S STUDENT. 
 
 More than 2,000 years have rolled away since the entire 
 Elements were given to the world. A halo of interest, 
 peculiar to no other work known among men, surrounds it. 
 It arose in the morning of Science, flourished among the 
 glorious Geometers of old, kept alive the sacred fire of 
 Learning, when almost the whole earth was without one 
 genial ray, and it now shines with undiminished lustre amid 
 the splendour of modern discovery. It has resisted the 
 attacks of countless critics, great and clever men—has sup¬ 
 ported the weight of innumerable commentators ; stiil it 
 remains the most perfect work of man, and the finest monu¬ 
 ment of his reasoning powers. It has been translated into 
 all languages, has been taught for centuries in every mathe- 
 
 Ritchie’s Principles of Geometry. 
 
INTRODUCTION TO Till-: 
 
 58 
 
 matical 6chool of emi/jence, and is now reputed the best 
 introduction to that sci ence extant. 
 
 It may safely be asserted, that as Grammar is the key ol 
 literature, so is Geometry the key of the sciences. Without 
 exaggeration, it has been contended that ‘ the only road to 
 accurate and independent knowledge of most of the” sciences 
 lies through the study of Mathematics.’* Euclid is the 
 great schoolmaster of the sciences. Ilis Elements are the 
 pass-word into their dominions. ‘ Without a sufficient 
 knowledge of Mathematics, that great instrument of [all] 
 exact inquiry, no man can ever make such advances in the 
 higher departments of science as can entitle him to form 
 an independent opinion on anv subject within their range.’f 
 
 Amasis wondered at the skill of Thales when he mea¬ 
 sured the Pyramids by means of their shadow. Neither the 
 power nor station of Amasis could place him on a level 
 with the distinguished geometer. Ptolemy, another Egyptian 
 king, is said to have asked Euclid if there was no easier 
 mode of becoming a geometrician than by studying his 
 Elements. 1 There is no royal road,' was his memorable 
 answer. The meaning of which is, that the acquisition of 
 no single idea can be made in a royal way. Only one path 
 was open to Ptolemy and to mankind. Science owns no 
 idle votaries. The only condition on which it yields its 
 secrets is that of attention. 
 
 Do not fall into the error of vanity, and neglect learning 
 under the impression that it courts you. It does indeed in¬ 
 vite you, but it is in the language of independence. It is a 
 benefactor, and never descends to supplicate the accept¬ 
 ance of its favours. It bestows honour and power on those 
 who seek its wealth, but it never yet conferred a gift on any 
 but the earnest. 
 
 Revolve the last words of the venerable professor of 
 Mathematics (Olinthus Gregory)—‘In your mental and 
 scientific pursuits; define to yourselves clearly the purposes 
 which you have in view; see to it that they are in no 
 way incompatible with the nature and the duty of man, and 
 you cannot direct your aim at too high a point.’ 
 
 Review for the Many. 
 
 ■f Sir John Herschel. 
 

 EUCLID. 
 
 .V.) 
 
 THE FIRST BOOK OF EUCLID. 
 
 SECTION I. 
 
 DEFINITIONS. 
 
 1. A Point is that which lias no parts, or which lias no 
 
 magnitude.* 
 
 2. A Line is length without brciulth.t 
 Corollary. —The extremities of a line are Points: and 
 
 the intersection of one line with another ia also a Point. 
 [Thus (fig. 4) the place A is a point]. 
 
 3. Right or Straight lines are such that cannot coincide 
 in any two points without coinciding altogether.} | This in 
 substituted for Euclid's, that the Corollary may appear 
 more evident]. 
 
 Cor. —Hence two straight lines cannot enclose a space. 
 Neither can two straight lines have a common segment 
 that is, they cannot coincide in part without coinciding 
 altogether. [The 14th prop, will illustrate this]. 
 
 * A Point is that which has no parts.- Dechairs. 
 
 A Point is that which has position, but not magnitude. 
 Playfair. 
 
 A Point is the extremity of a lino having no dimension* of any 
 kind—neither length, nor breadth, nor thickness.- (team. Lth. 
 Useful Knowledge. 
 
 A Point is a mark in magnitude, which is (supposed to b<; 
 
 Indivisible.— Zeno. 
 
 A Point is the beginning, as it were, of all magnitude, as unity 
 
 is of number.— Parquet. 
 
 A Point occupies no space. — Legendre. 
 
 t A Line has length only, and wants all breadth, Inasmuch a* 
 it is understood to be produced by the flowing of a point.- Parquet. 
 
 I A Right line is that whose extremes hide all the rest.— Plato. 
 A Right line is the shortest of all those that can be dr a *r» 
 between two points.— ArclarnerJet. 
 
GO 
 
 T.VCLiD. 
 
 4. Parallel straight lines are such as are in the same 
 plane, and which being produced ever so far both ways do 
 not meet.* * (See fig. 5). 
 
 5. A Superficies , or surface, i3 that which has only 
 length and breadth. 
 
 Cor.—T he extremities of a superficies are lines, and 
 the intersections of one superficies with another are also 
 lines.+ 
 
 G. A Plane , or plane surface, is that in which any two 
 points being taken, the straight line between them lies 
 wholly in that superficies.? 
 
 7. A Plane Angle is the inclination of two lines to one 
 another, which meet together, but are not in the same right 
 line. 
 
 [The two lines, which, meeting together, make an 
 angle, are called the legs of that angle. A B, A C, (fig. 7) 
 arc the legs of the angle B A C. The point A, at which 
 the legs meet, is called the vertex of the angle. An angle 
 may be designated by a single letter, when its legs are the 
 only lines which meet at its vertex. But when more than 
 two lines meet at the same point, in order to avoid confusion 
 three letters are employed to designate the augle at that 
 point—the letter which marks the vertex of the angle being 
 always placed in the middle. Thus (fig. 6) the lines G C 
 and C meeting together at C, make the angle G C E, or 
 E C G: the lines G C, E C, are the legs of the angle, the 
 
 * Lines are Parallel to each other if the perpendiculars be 
 tween them are all equal to each other— Archimedes. 
 
 Newton, in Lemma 22, Bk. I. of his ‘ Principia,’ says that 
 parallels are such lines as tend to a point infinitely distant. 
 
 f A Superficies has two dimensions, and is understood to be 
 produced by the flowing of a line.— Tacquet. 
 
 • A Plane, surface is a surface which is perpetually even, or 
 fiat throughout its whole extent— Darlei/. 
 
 (On such planes all the figures in the First Book are supposed 
 to be constructed). 
 
 A Sulid is a.magnitude having the three dimensions of space : 
 length, breadth, and depth. Any one of the six sides of a Cube 
 is a surface, or superficies — an edge is a line —a corner a point. 
 (See fig. 30). 
 
EUCLID. 
 
 61 
 
 point C is the vci'tex. In like manner may be named the 
 angle E C A, which is the sum of the angles G C E, GCA: 
 and so of the other angles round the same point. 
 
 When the legs of an angle are produced (or continued) 
 beyond the vertex, the angles made by them on both sides 
 of the vertex are said to be vertically opposite to each other: 
 thus, since G C is continued to D, and E C to F, the angles 
 G C E, D C F, are said to be vertically opposite to each 
 other. And so of the other angles on each side C. 
 
 It must be understood, that by an angle is not meant 
 the surface between the lines which form it. For 
 though the surface be increased by producing the legs, (to 
 D and E, for instance, in fig. 7), the angle will still remain 
 the same in magnitude. By an angle , in fact, is meant the 
 degree of width, or separation between the lines that form 
 it: thus, the opening between B A C is greater than the 
 angle F A C.] 
 
 8. When a straight line, standing on another straight 
 line, makes the adjacent angles equal to each other, each of 
 these angles is called a Right Angle , and each of these lines 
 is said to be at right angles to, or perpendicular to the 
 
 other.* (See fig. 8). 
 
 9. An Obtuse angle is that which is greater than a right 
 angle. (See angle CBA, fig. 49). 
 
 10. An Acute angle is that which is less than a right 
 angle. (See angle A B D, fig. 49). 
 
 11. A Figure is that which is bounded by one or more 
 lines. (See fig. 9). The space enclosed within a figure is 
 called its area. 
 
 12. A Circle is a plane figure bounded by one line 
 called the circumference , or periphery; of which all lines 
 drawn from a certain point within the figure to the circum¬ 
 ference are equal to one another.-! (See fig. 10). 
 
 • A carpenter’s square is a ‘ right angle.’ Pythagoras is said 
 to have been the inventor of the Square. 
 
 A perpendicular line inclines neither to the right hand nor to 
 the left. From this circumspect idea probably arose the pre¬ 
 cept of the poet— 
 
 ‘ Far from extremes a middle course is best.’ 
 
 f Till the time of Plato this was the only curve admitted into 
 
 Geometry. 
 
EUCLID. 
 
 <52 
 
 13. That point is called the Centre of the circle. 
 
 14. A Diameter of a circle is a straight line drawn 
 through the centre, and terminating on both sides iD the 
 circumference. 
 
 15. A line drawn from the centre to the circumference 
 of a circle is called the Radius , or Semi-diameter. 
 
 [Radii is the plural of radius]. 
 
 1G. Rectilineal figures are those contained by straight 
 lines. 
 
 17. Trilateral figures, or Triangles, by three straight 
 lines. (See fig. 11). 
 
 18. Quadrilateral figures by four straight lines. (See 
 fig. 12). 
 
 19. Of three-sided figures: an Equilateral triangle is 
 that which has three equal sides. (See triangle A C B, 
 fig. 31). 
 
 20. An Isosceles triangle is that which has two equal 
 sides. (See triangle A B C, fig. 37). 
 
 21. A Right-angled triangle is that which has a right 
 angle.* See triangle A E 1), fig. 136.) 
 
 [The angle opposite the base is called the vertical angle. 
 A side of a triangle, considered in reference to the angle 
 opposite, is said to subtend it, or be the subtense of the 
 augle. When the side of any triangle is produced, as B C 
 to D, (fig. 14) the angle A C 1), made by the produced part 
 C D, with the other leg A C, is called the external angle. 
 And the angle A C B, adjacent to the external angle, is 
 called the internal adjacent angle. The other tw T o angles of 
 the triangle are together called the internal remote angles: 
 and of these, that of which the produced side is a leg, 
 namely, ABC, is the internal opposite , and the other, 
 namely, B A C, is the internal alternate angle.] 
 
 22. Of quadrilateral figures : a parallelogram is that of 
 which the opposite sides are parallel. (See fig. 15). 
 
 23. The straight line which joins the opposite angles of 
 a quadrilateral figure is called a Diagonal. 
 
 * Any side of a rectilineal figure may be called the Base. In 
 a right-angled triangle (see fig. 13) the side a c, opposite the 
 right angle is called the Hypothenense, either of the other two sides 
 a b, a c, the Base, and the one not taken the perpendicular. 
 
EUCLID. 
 
 63 
 
 24. The Altitude of a triangle or a parallelogram is a 
 perpendicular drawn from the opposite angle or side, upon 
 the base, as a b, c d. (See figs. 1G and 17.) 
 
 25. Of parallelograms: a Square is that which has all its 
 sides and angles equal. (See fig. 97). 
 
 26. An Obloyig, or Rectangle is a four-sided figure, that 
 has equal angles but not equal sides. (See fig. 23). [When 
 a four-sided figure lias equal angles, they are always right 
 angles. Every * square ’ is a ‘ rectangle,’ but every * rect¬ 
 angle ’ is not a square.’] 
 
 SECTION II 
 
 POSTULATES. 
 
 (Things required : from the Latin postulo, to require). 
 
 1. * Let it be granted that a straight line may be drawn 
 from any one point to any other point. 
 
 2. That a terminated straight line may be produced to 
 any length in a straight line. 
 
 3. And that a circle may be described from any centre 
 with any radius or interval. 
 
 SECTION III. 
 
 AXIOMS. 
 
 (Authorities, or things haring authority : from a Greek word. 
 An Axiom is a self-evident proposition). 
 
 1. Things which are equal to the same things, arc equal 
 
 to one another. 
 
 2. If equals be added to equals, the wholes are equal. 
 
 3. If equals be taken from equals, the remainders are 
 
 equal. 
 
 4. If equals be added to unequals, the sums are unequal. 
 
 5. If equals be taken from unequals, the remainders are 
 
 unequal. 
 
EUCLID. 
 
 GI 
 
 0. Things which are doubles of the same thing, are 
 equal to one another. 
 
 7. Things which are halves of the same thing, are equa. 
 to one another. 
 
 8. Magnitudes which coincide with one another, (that 
 is to say, which fit together so exactly, that every point ot 
 the one lies on some point of the other,) are equal. 
 
 9. The whole is greater than its part. 
 
 10. All right angles are equal to one another. (Legen¬ 
 dre demonstrates this). 
 
 11. Two straight lines which intersect each other can¬ 
 not be both parallel to the same straight line. 
 
 [All these Axioms arc not required in the First Book, 
 but not being numerous they are inserted as samples of the 
 assumptions of Geometry]. 
 
 SECTION IV. 
 
 DEFINITIONS OF TERMS. 
 
 A Theorem is a truth which becomes evident by a train of 
 reasoning called a Demonstration. 
 
 A Problem is a question proposed which requires a 
 solution. 
 
 A Lemma is a subordinate or minor truth, which is estab¬ 
 lished in order to be employed in the demonstration of a 
 theorem, or in the solution of a problem. 
 
 A Proposition is a portion of science. It is a common 
 name applied indifferently to theorems, problems, and 
 lemmas. A proposition is a simple statement respecting 
 any subject.— Jas. Milne. 
 
 A Hypothesis is a fact assumed without proof. (Thus, 
 when it is affirmed that in an Isosceles triangle the angles 
 at the base are equal, the hypothesis of the proposition is 
 that the triangle is Isosceles, or that its legs are equal). 
 
 A Construction is the addition made to, or change made 
 in, the original figure, by dividing or drawing lines, in 
 order to adapt it to the argument of the demonstration, or 
 the solution of the problem. The condition under which 
 
EUCLID. G5 
 
 these changes arc made is as indisputable as those contained 
 in the hypothesis. Thus, if a line be made equal to a given 
 line, these two lines are said to be equal by construction. 
 
 A Corollary is a consequence easily deduced from one or 
 more propositions. 
 
 A Scholium is a remark on one or more propositions, 
 which explains the application, connection, . limitation, 
 extension, or other important circumstance belonging thereto. 
 
 A Demonstration is a process or reasoning. It is either 
 direct or indirect. 
 
 [To demonstrate a proposition is to exhibit a sufficient 
 reason for the action affirmed by that proposition.— Jas. 
 Milne.'] 
 
 A Direct demonstration is a regular process of reasoning, 
 from the premises to the conclusion. 
 
 An Indirect demonstration establishes a proposition, by- 
 proving that any hypothesis contrary to it is contradictory, 
 or absurd : and it is therefore sometimes called a reductio 
 ad absurdum. 
 
 The Data or Premises of a proposition are the relations 
 or conditions, granted or given, from which now relations 
 are to be reasoned out, or a construction to be effected. 
 
 To Bisect is to divide into two equal parts. 
 
 To Trisect , to divide into three equal parts. 
 
 An Enunciation declares what is to be proved or done. 
 
 The Conclusion asserts that the thing required has been 
 proved or done. 
 
 SECTION V. 
 
 EXPLANATIONS OF SIGNS. 
 
 The arguments employed to demonstrate any proposition 
 in Geometry must be founded more or less on the antecedent 
 Propositions, Corollaries, or on the Definitions, Postulates, 
 Axioms, Hypotheses, and Constructions. These are res¬ 
 pectively referred to, in the following pages, under the 
 abridged forms Prop., Cor., Dcf., Post., Ax., Hyp., Const. 
 
 The text of Dr. Simpson, used in the Universities, did him 
 infinite credit in his day : but language has advanced since, 
 
GO 
 
 EUCLID. 
 
 and the amplifications and reiterations he employs are 
 needed now only in oral discourses. Mr. Cooley appears 
 to be the most successful of all modern editors of Euclid in 
 adopting a style of superior brevity and.clearness, and has 
 thus effected an important abridgement without omitting a 
 single step in the reasoning, or in the slightest degree im¬ 
 pairing its strength and validity. Iiis text is that chiefly 
 followed in the succeeding section. 
 
 SECTION VI. 
 
 Mi - BOR&MO AND rnOBCEM3. 
 
 PRELIMINARY DIRECTIONS. 
 
 1. Remember that in Geometry all sections, paragraphs, and 
 sentences are of nearly equal importance. 
 
 2. Read deliberately, every paragraph, that you fully under¬ 
 stand it. 
 
 ‘ Learn to read sloirli /, all other graces 
 Will follow in their proper places 
 is an all essential injunction to the young elocutionist, and 
 equally applicable to the young geometrician. 
 
 3. Note carefully the parts of every proposition. 
 
 4. Make sure of the true meaning of every term you meet 
 with. 
 
 5. Trace every reference. 
 
 6. Study each proposition until you can prove it yourself 
 iii.t/ioul the hook—till the principle of the demonstration is fa¬ 
 miliarised to the understanding. 
 
 7. The Plates of diagrams are intended only to give initiative 
 ideas, and not to obviate the necessity of making new figures. 
 The learner will best understand that which he 3 has made himself. 
 G eometry is to be. acquired best by practice. The student should 
 in every case construct his own diagram according to the * con- 
 s 1 ruction iu each proposition, making it as large as convenient, 
 as largeness of diagram assists the conception of its purport. ; 
 
 8. Students perpetually ask—Must the Definitions arid Axioms 
 be committed to memory ? Not necessarily so. It matters little 
 that the memory* is taxed if the understanding is excited. Let 
 the learner make the references to the Definitions and Axioms as 
 often as they are mentioned— 1 which will be at the precise moment 
 
PREFACE. 
 
 This work was originally prepared at the request of a 
 Society which took great interest in popular educa¬ 
 tion. Several things, however, interfered to prevent 
 that body from publishing it, and it was returned to 
 me with kind words of approval and regret. Since 
 then, I have waited for an occasion of presenting it 
 myself to the service and judgment of the public. 
 
 Possibly, this delay may prove an advantage to the 
 book, since it has afforded me the opportunity of revis¬ 
 ing it, when the first fondness of authorship had sub¬ 
 sided into the more solid ambition of writing for the 
 improvement of others. 
 
 The criticism this confession may invoke, I shall 
 not be found to deprecate, but shall keep in rccollee 
 tion the prudent advice of Helicane, and 
 
 With patience bear 
 
 Such griefs as I do lay upon myself. 
 
 My aims have been:—1. Tn pgeaeribe the limits of JPCSCH 
 Mathematical learning, defining w'hat must be acquired 
 and what may be neglected. 2. The di&cussion of the 
 supposed connection between Arithmetic and Mathe¬ 
 matics, and explanation of certain important distinctions 
 generally confined to the class room. 3. Presenting 
 a history of the rise and progress of Mathematics, 
 somewhat more complete than previous ones. 4. Ex¬ 
 tending the view of the utility of Mathematics as a 
 means of gauging the physical world. 5. A de- 
 
 A 
 
4 
 
 PREFACE. 
 
 velopment of the argument in favour of this science 
 as a means of mental discipline, guarded from the 
 arrogance and exaggeration which have so often des¬ 
 troyed its efficiency. 6. An exposition of Mathe- 
 matical logic, and the true principles of Science, 
 illustrating the value of the one and the simplicity 
 of the other. 7. The enforcement of Natural Geo¬ 
 metry—resting it on the foundation of the common 
 understanding, distinguishing its Beauties and Uses, 
 adapting it to the wants of the many, to the capacities 
 of the young, to the aptitudes of the uninitiated, and 
 the exigencies of men of business—the people of little 
 time and a definite purpose. 
 
 G. J. H. 
 
 lliT The word euclid, on the title page, contains 
 the Diagrams of six of the most interesting and cele¬ 
 brated Propositions of Euclid. The number of each 
 Proposition, and the number of the Book in which it 
 is found, are placed at the foot of each letter. The 
 Diagrams in e u c occurring in the First Book, 
 the reader will discover, in the following pages, to 
 what they relate. The Diagram in l is employed 
 to prove that all the angles that can be made in a 
 semicircle are right angles. The Diagram in i be¬ 
 longs to the Proposition respecting the proportion 
 between triangles and parallelograms of the same alti¬ 
 tude. The Diagram in d is the geometrical founda¬ 
 tion of the Rule of Three. It seemed the most ap¬ 
 propriate tribute to Euclid to associate his works with 
 his name. 
 
CONTENTS. 
 
 INTRODUCTION TO THE MATHEMATICS. 
 
 Chap. 1.—Discouraging Influence of a certain popular Maxim 
 over the Pursuit of Learning in general, and Mathematics 
 in particular. 
 
 Chap. 2.—Important Distinctions pertaining to Mathematics. 
 Chap. 3.—History of the Rise and Progress of the Mathe¬ 
 matics. 
 
 Chap. 4.—Utility of Mathematics as a means of extending 
 our knowledge of the physical world. 
 
 Chap. 5.—Mathematics as a means of Mental Discipline. 
 Chap. G.—The Logic of Euclid. 
 
 Chap. 7-—Natural Geometry. 
 
 Chap. 8.—Practice and Theory, or the Distinction between 
 Practical Geometry and Pure Mathematics. 
 
 Chap. 9.—Exordial Address to the Student. 
 
 THE FIRST BOOK OF EUCLID. 
 
 Sect. 1.—Definitions. 
 
 Sect. 2.—Postulates. 
 
 Sect. 3.—Axioms. 
 
 Sect. 4.—Definition of Terms. 
 
 Sect. 5.—Explanations. 
 
 Sect. 6.—Preliminary Directions. 
 
 Sect. 7.—Problems and Theorems. 
 
 Sect. 8.—Exercises. 
 
 Sect. 9.—Supplementary Illustrations of Geometrical Logic. 
 USES OF EUCLID. 
 
 Chap. 1.—The Practical Uses of the Definitions. 
 
 Chap. 2.—Miscellaneous Definitions, introductory to the Uses 
 of the Propositions. 
 
 Chap. 3.—Uses of the Problems and Theorems of Euclid. 
 
INTRODUCTION TO THE MATHEMATICS. ' 
 
 CHAPTER I. 
 
 DISCOURAGING INFLUENCE OF A CERTAIN POPULAR 
 MAXIM OVER THE PURSUIT OF LEARNING IN GENE¬ 
 RAL, AND MATHEMATICS IN PARTICULAR. 
 
 ( Whatever is worth while doing at all, it is worth 
 while doing well/ is a maxim of deserved reputation: 
 and when applied in its proper sense, of great service. 
 But, adopted as this has been, without due regard to 
 its exact import, it may be questioned whether any 
 maxim ever proved so serious a discouragement to the 
 majority of learners. "What it is worth whfle to do at all, 
 it is, unquestionably, sound wisdom and true economy 
 to do well. But what it is worth while to perform, 
 and the measure of the undertaking, are to be decided 
 by the wants and circumstances of the person engag¬ 
 ing in the pursuit. Instead of this being done we 
 have those who, on feeling interested in the investi¬ 
 gation of any branch of learning, conclude it to be 
 incumbent on them to entirely explore it, whatever be 
 its nature or extent, and, wanting time to accomplish 
 this, often an extravagant task, remain in entire 
 ignorance of that of which it was probably important 
 they should know .something. 
 
 The person destined for the profession of Chemistry, 
 of necessity must acquaint himself with the works of 
 Priestley, Davy, Dalton, Liebig, Faraday, Daniels, 
 and other eminent men in that department of know¬ 
 ledge : but the general inquirer will sufficiently in- 
 
8 
 
 INTRODUCTION TO THE 
 
 form himself on this subject from Chambers’ ‘ Rudi¬ 
 ments ’ of the science. In like manner, a person 
 intended for the Medical profession has a wide class 
 of authors before him, with whom he must become 
 conversant, while the artizan would be considered well 
 informed who was familiar with the works of South- 
 wood Smith and Andrew Combe. Now, what it is 
 proper for the Chemist and Medical student to under¬ 
 take, is no less than the whole range of knowledge 
 belonging to each of their respective professions—and 
 this it is worth their while to master well. But the 
 general inquirer and the artizan, being differently cir¬ 
 cumstanced, require only such a general insight into 
 these subjects as shall enable them to converse thereon 
 with propriety, to understand books in which they 
 are introduced, an to apply the plain and popular 
 rules to the busin s, enjoyment, and preservation of 
 life. 
 
 The judicious rule of acquisition is this. After mas¬ 
 tering the knowledge proper to individual profession, 
 we next have to seek general knowledge: for he 
 who should essay to sound the depths of every sub¬ 
 ject, would find life exhausted before he had com¬ 
 pleted a tithe of his task. A course which would 
 condemn to the unremitting toil of hoarding infor¬ 
 mation, and exclude for ever from the pleasure of 
 its application. 
 
 Unmindful of considerations of this kind, it is com¬ 
 monly thought, that whenever Geometry is proposed 
 for investigation, (since whatever is worth while 
 doing at all is worth while doing well), nothing less 
 than Dr. Simpson’s Eight Books of Euclid, his name¬ 
 sake’s Trigonometry, Bonnycastle’s Mensuration, 
 Bridge’s Algebra, and a host of scientific treatises, 
 must necessarily be entered upon. And as it is only 
 the time of professed students that will allow of this, the 
 
VAi Q-e^yitr-o.C j ^tujmrCV' d-fi-t rtn ituj 
 W /vn^ixMiy lw>w4- (il_ J 
 
 “^attention, and earnestly devotes himself to that, and 
 to nothin^ else. His business is to explore, not to 
 conquer the region of Mathematics It is that which 
 it is worth his while to do, and he does that well. 
 
 Many people of undisciplined powers, and ignorant 
 of the art of acquiring knowledge, can never move 
 anywhere unless they move profoundly. Such people 
 soon lose themselves. Instead of being the masters of 
 
 knowledge, knowledge is the master of them. By 
 being profound out of place, they bring profundity 
 into contempt. It is not within the compass of human 
 power to be profound upon more than a few subjects 
 The poet was somewhat too strict in his limitation, 
 but he approached the truth in declaring— 
 
 Ono subject only will one fynius fit, 
 
 So vast is art—so narrow thman wit. 
 
 Do not be deterred from the useful course of general 
 investigation, in these two Sections recommended, by 
 the vague charge of superficiality,' so often preferred 
 without propriety and listened to without advantage. 
 Many persons live and die in disreputable ignorance 
 because they are determined not to be superficial. 
 Superficiality is a serious imputation when it respects 
 what you ought to know well. But with respect to 
 - general knowledge, it is no more criminal to be super¬ 
 ficial than to have only five senses, or not to be in half 
 a dozen places at once. It is the condition of humanity 
 only to be able to take a general survey of the wide 
 field of knowledge. The republic of literature lias no 
 inapt resemblance to the state of good citizenship, in 
 which a man is expected to know his own business, 
 and to have such an acquaintance only with that of 
 his neighbours as shall give him a laudable interest in 
 their pursuits, and assure him that they are conducive 
 to the dignity of the commonwealth. 
 
■ troance t 
 for wliic 
 
 [5 rcaaerlo the History of the Mathematics, 
 i he is prepared. The record of its gra’ 
 
 - nUTniraur : 
 
 —Otmven'aiions urrtimi suDjeec, lu<f rcqtnsiu; 
 
 inform ati >n—the First Book of Euclid will srccomplish 
 the rest. It will introduce him to both 1 heoretical 
 and Prac ical Geometry, and to the technic d charac¬ 
 teristics ot Mathematics. It will enable him to under¬ 
 stand the modus operandi of Newton’s Prir cipia. It 
 will famili arise him with the illustrations of Dynamics 
 and Mech mics. After performing the 1 Com tructions’ 
 of Euclid, and putting them to the 1 Uses ’ £ t the end 
 of this boc k, any person could understand an<. practice 
 Keith’s p ’inciples of Mapping, and illus rate his 
 geography by constructing his own maps. We have 
 seen in tl e quotation in the last chapter, Ifrom the 
 Saturday , Tournal, how the First Book of 
 it does not make the young pupil master of As 
 it will place him on the same common groi 
 the Astronomer himself, and familiarise him 
 scientific skerets of that fine pursuit. 
 
 First Book of Euclid does not make the 
 Mathematician, it so qualifies him that he 
 Mathematician whenever he pleases, as it i 
 
 clear insight of that science which will enable him to 
 
 uclid, if 
 ronomy, 
 nd with 
 with the 
 Indec d, if the 
 tudent a 
 ;an be a 
 n parts a 
 
[ 
 
 'nr/* Ao^o^ A t /^e f<n</ *i- —• 
 
 V^r^ouj / ^ r ^ ^ ^ ~ff^, 
 
 * # ' S /As /. Sst'l s si //Asi /'d— /WsfAS iZ 
 
 Au^'i > S<r 
 
 M/turUf** **-<**-** 
 
 II fit, CfctreJl*'^ /t^t- 
 
 'k /u</«<'&' — A' 
 
 r/ w ^w A 
 

 
MATHEMATICS. 
 
 9 
 
 entire subject is generally neglected.* In what be 
 longs to a man’s profession, bis duty, or his chosen 
 study, there must be no superficialness. The ])ro- 
 fessional astronomer must be master of Newton—the 
 mariner, of Thomas Simpson—the architect and civil 
 engineer, of Euler and the Bernouillis, and a certain 
 class of able geometricians. So much of geometry as 
 may belong to any branch of mechanics, must be mas¬ 
 tered by all who would be proficients in such branch 
 —whatever be the formidable array of works to be 
 encountered. But under other circumstances, what 
 is chiefly wanted is such a general acquaintance with 
 the subject as shall enable a person to distinctly under¬ 
 stand the nature and application of Mathematics—the 
 process of geometrical reasoning—the meaning of the 
 technical terms now so frequent in the scientific lec¬ 
 ture room, and in treatises on Mechanics.f 
 
 * Of forty artizans who commenced tho study of Mathe¬ 
 matics, under this impression, in the Birmingham Mechanics’ 
 Institute (1837-9) scarcely one-eighth pursued tho subject 
 to a satisfactory issue: and the few who did, pursuing 
 it at the usual length, had to make large abstractions from 
 the hours of recreation and rest, and, to a great extent, paid 
 their health as the penalty of their application. 
 
 + Greek and Latin, Aristotle’s logic and classical versifica¬ 
 tion, quadratic equations, conic sections, the differential calcu¬ 
 lus, are very good things.—They are essential to the fame of a 
 Parr or a Porson, a Herschel or a Whewell. These acquisi¬ 
 tions arc, doubtless, amongst the greatest triumphs of tho 
 human understanding, and are calculated to raise a few—per¬ 
 haps one in a hundred—to distinction ; but upon the minds 
 of the remaining ninety-nine they produce no sort of impres¬ 
 sion. They do not do much mischief to such persons in 
 themselves, but they are of incalculable detriment, and the 
 industry which they absorb to no available purpose. Ten 
 years of youth—the most valuable and important period of 
 life—are wasted in studies which, to nineteen-twentieths of 
 the persons engaged in them, are of no use whatever in 
 future yesivs.—BlacktroocTs Magazine, Nov. 1845. 
 
10 
 
 INTUO’DTJ'CTION TO TTTE 
 
 Much has been said of the qualities of calmness 1 
 of clearness, coherency, and cogency—the reputed 
 traits of a geometrical mind : and a general de- I 
 sire of acquaintance with the writings of Euclid is 
 felt on this account. Now, these points enumerated ! 
 may be learned in a very satisfactory manner from 
 the First Book of Euclid, which six weeks’ application 1 
 will master. On this account the First Book only is 
 introduced in this Work — that Book constituting 1 
 a complete introduction to geometrical science. 
 
 The usual works on ‘ Geometry,’ and those under the ; 
 name of ‘ Euclid’s Elements,’ have been mostly intended j 
 for professional persons, or teachers, as few besides ] 
 were likely to buy them. Sometimes they have been I 
 issued by individuals who merely sought geometrical I 
 reputation. In both cases such works would natur¬ 
 ally contain much which the general student would j 
 never want. Now that enquirers have multiplied, j 
 the pedantry of the schools has been brushed away 
 in the bustle of practical life, and zeal of learning 
 broken through the barriers of caste — the popu¬ 
 lace claim to have opened to them the portals of 
 Geometry. Euclid, by the unanimous consent of 
 learned men, is the keeper of the gate. His First 
 Book is the entrance to the Temple—and it is strange 
 that none among the moderns have selected this vesti¬ 
 bule for the admission of the public to view the 
 museum of the exact sciences. 
 
 Proclus, the last Greek geometer, seems to have 
 had this idea. He alone, of the ancients, is recorded 
 to have paid distinguished attention to the First Book 
 of Euclid. He left a special commentary upon it, 
 abounding in curious observations on the metaphysics 
 of Euclid, and with curious facts connected with the 
 science. 
 
 The desirability of recasting our educational trea- 
 
MATHEMATICS. 
 
 11 
 
 tises the commercial habits of the age leave be¬ 
 yond doubt. The industrious classes are they 'who 
 most need the acquisition of any logical power which 
 Geometry can impart, and their means of leisure to 
 pursue such a study are few and far between. Ac¬ 
 tuated, probably, by convictions of this kind, Sir John 
 Herschel has composed a work on Astronomy, in¬ 
 tended to answer for the astronomical, such purposes as 
 are here contemplated for the geometrical public. 
 Indeed, many distinguished men have turned their 
 attention to the same object in their respective sciences. 
 Of the fitness of the single book of Euclid for the 
 accomplishment of the ends here specially in view, 
 the reader will be able to judge from the exe¬ 
 cution of these pages. Since the writer first arranged 
 them, a coincident opinion has been very distinctly 
 expressed upon the subject.— 
 
 ‘ To young readers, about to enter on the study of 
 Astronomy, and who seek only to get such a mere 
 general knowledge of it as may satisfy their own 
 minds, we would say in the first place, acquire a 
 knowledge, however slight it may be, of the elements 
 of Mathematics. Your mind may not be of a mathe¬ 
 matical turn, and there may not be the slightest pros¬ 
 pect of your deriving any positive advantage from 
 posing your brains with the “ First Book of Euclid .” 
 No matter; try and go over it; it is worth your 
 while. You cannot stir in Astronomy without know¬ 
 ing something of the properties of the circle and tri¬ 
 angle. He, therefore, who wishes to comprehend the 
 “ reasons ” on which Astronomy is based, will acquire 
 this preliminary knowledge, without which it is useless 
 for him to enter upon the study. After he has ac¬ 
 quired it, and after he has studied an Astronomical 
 work, he may be far—very far, indeed, from having 
 the smallest pretension to the name of Astronomer, 
 
12 
 
 INTRODUCTION, LTC. 
 
 but he will be in possession of a few of the “ funda* 1 
 mentals ” of the science; and will stand on the same 
 platform with the Astronomer himself.’* 
 
 The world has been favoured with the Beauties of 3 
 Beattie, of Byron, Scott, and of its' great benefactors in I 
 literature and song, and the public who could never l 
 acquaint themselves with the entire productions of i 
 these authors, have become familiar, through such con- 
 venient epitomes, with their merits and their graces. j 
 That there are both romance and beauty of a stem kind 1 
 in the wonderful writings of Euclid, no man of infor- i 
 mation or taste will deny. His works were the torch ’ 
 which lighted the sciences to their earliest discoveries. ] 
 They revealed to us the world of form and the 
 hidden secrets of figure, and. enlisted lines and 
 angles in the eternal service of man. The deep en¬ 
 thusiasm which some have felt for Geometiy will 
 be generally shared in the current diffusion of 
 intelligence. An important connection is beginning 
 to be established between the principles of science and 
 morality. More than one philosoper is found to ac¬ 
 knowledge, that ‘The axioms of Physics translate 
 those of Ethics.’f The desire of aiding to some extent 
 the development of this fact was not one of the least 
 reasons which has dictated the compilation of the 
 ‘ Beauties and Uses of Euclid.’ 
 
 London Saturday Journal, 
 t Emerson. 
 
INTRODUCTION TO THE MATHEMATICS. 
 
 CHAPTER I. 
 
 DISCOURAGING INFLUENCE OF A CERTAIN POPULAR 
 MAXIM OVER THE PURSUIT OF LEARNING IN GENE¬ 
 RAL, AND MATHEMATICS IN PARTICULAR. 
 
 ‘Whatever is worth while doing at all, it is worth 
 while doing well,' is a maxim of deserved reputation: 
 and when applied in its proper sense, of great service. 
 But, adopted as this has been, without due regard to 
 its exact import, it may be questioned whether any 
 maxim ever proved so serious a discouragement to the 
 majority of learners. What it is worth while to do at all, 
 it is, unquestionably, sound wisdom and true economy 
 to do well. But v'hat it is worth while to perform, 
 and the measure of the undertaking, are to be decided 
 by the wants and circumstances of the person engag¬ 
 ing in the pursuit. Instead of this being done we 
 have those who, on feeling interested in the investi¬ 
 gation of any branch of learning, conclude it to be 
 incumbent on them to entirely explore it, whatever be 
 its nature or extent, and, wanting time to accomplish 
 this, often an extravagant task, remain in entire 
 ignorance of that of which it was probably important 
 tney should know .something. 
 
 The person destined for the profession of Chemistry, 
 of necessity must acquaint himself with the works of 
 Priestley, Davy, Dalton, Liebig, Faraday, Daniels, 
 and other eminent men in that department ot know¬ 
 ledge : but the general inquirer will sufficiently in- 
 
8 
 
 INTRODUCTION TO THE 
 
 form himself on this subject from Chambers’«Rudi- j 
 ments’ of the science. In like manner, a person 
 intended for the Medical profession has a wide class 
 of authors before him, with whom he must become 
 conversant, while the artizan would be considered well 
 informed who was familiar with the works of South- 
 wood Smith and Andrew Combe. Now, what it is 
 proper for the Chemist and Medical student to under¬ 
 take, is no less than the whole range of knowledge 
 belonging to each of their respective professions—and 
 this it is worth their while to master w'ell. But the 
 general inquirer and the artizan, being differently cir¬ 
 cumstanced, require only such a general insight into 
 these subjects as shall enable them to converse thereon 
 with propriety, to understand books in which they 
 are introduced, and to apply the plain and popular 
 rules to the business, enjoyment, and preservation of 
 life. 
 
 The judicious rule of acquisition is this. After mas¬ 
 tering the knowledge proper to individual profession, 
 we next have to seek general knowledge: for he 
 who should essay to sound the depths of every sub¬ 
 ject, would find life exhausted before he had com¬ 
 pleted a tithe of his task. A course which would 
 condemn to the unremitting toil of hoarding infor¬ 
 mation, and exclude for ever from the pleasure of 
 its application. 
 
 Unmindful of considerations of this kind, it is com¬ 
 monly thought, that whenever Geometry is proposed 
 for investigation, (since whatever is worth while 
 doing at all is worth while doing well), nothing less 
 than Dr. Simpson’s Eight Books of Euclid, his name¬ 
 sake’s Trigonometry, Bonny castle’s Mensuration, 
 Bridge’s Algebra, and a host of scientific treatises, 
 must necessarily be entered upon. And as it is only 
 the time of professed students that will allow of this, the 
 
MATHEMATICS. 
 
 9 
 
 entire subject is generally neglected.* In what be 
 lon^s to a man’s profession, bis duty, or his chosen 
 etudy, there must be no superficialness. The pro¬ 
 fessional astronomer must be master of Newton—the 
 mariner, of Thomas Simpson—the architect and civil 
 engineer, of Euler and the Bernouillis, and a certain 
 class of able geometricians. So much of geometry as 
 may belong to any branch of mechanics, must be mas¬ 
 tered by all who would be proficients in such branch 
 
 _whatever be the formidable array of works to be 
 
 encountered. But under other circumstances, what 
 is chiefly wanted is such a general acquaintance with 
 the subject as shall enable a person to distinctly under¬ 
 stand the nature and application of Mathematics—the 
 process of geometrical reasoning—the meaning of the 
 technical terms now so frequent in the scientific lec¬ 
 ture room, and in treatises on Mechanics.! 
 
 * Of forty artizans who commenced the study of Mathe¬ 
 matics, under this impression, in the Birmingham Mechanics’ 
 Institute (1837-9) scarcely one-eighth pursued the subject 
 to a satisfactory issue: and the few who did, pursuing 
 it at the usual length, had to make large abstractions from 
 the hours of recreation and rest, and, to a great extent, paid 
 their health as the penalty of their application. 
 
 t Greek and Latin, Aristotle’s logic and classical versifica¬ 
 tion, quadratic equations, conic sections, the differential calcu¬ 
 lus, .are very good things.—They are essential to the fame of a 
 l’arr or a Porson, a Herschel or a Whewell. These acquisi¬ 
 tions are, doubtless, amongst the greatest triumphs of the 
 human understanding, and are calculated to raise a few-^-per- 
 haps one in a hundred—to distinction ; but upon the minds 
 of the remaining ninety-nine they produce no sort of impres¬ 
 sion. They do not do much mischief to such persons in 
 themselves, but they are of incalculable detriment, and the 
 industry which they absorb to no available purpose. Ten 
 years of youth—the most valuable and important period of 
 life—are wasted in studies which, to nineteen-twentieths of 
 the persons engaged in them, are of no use whatever in 
 future years .—BlackuoocTs Magazine, Nov. 1845. 
 
10 
 
 INTRODUCTION TO THE 
 
 Much has been said of the qualities of calmness 1 
 oi clearness, coherency, and cogency—the reputil 
 traits of a geometrical mind : and a general de- 
 sire of acquaintance with the writings of Euclid is 
 felt on this account. Now, these points enumerated - 
 may be learned in a very satisfactory manner from 
 the First Book of Euclid, which six weeks’ application 
 will master. On this account the First Book onlyij 
 introduced in this Work —that Book constituting 
 a complete introduction to geometrical science. .1 
 
 The usual works on ‘Geometry,’ and those under the 
 name of ‘ Euclid’s Elements,’ have been mostly intended 
 for professional persons, or teachers, as few besides 
 were likely to buy them. Sometimes they have been 
 issued by individuals who merely sought geometrical 
 reputation. In both cases such works would natur¬ 
 ally contain much which the general student would 
 never want. Now that enquirers have multiplied, 
 the pedantry of the schools has been brushed away 
 in the bustle of practical life, and zeal of learning 
 broken through tne barriers of caste — the popu¬ 
 lace claim to have opened to them the portals of 
 Geometry. Euclid, by the unanimous consent of 
 learned men, is the keeper of the gate. His First 
 Book is the entrance to the Temple—and it is strange 
 that none among the moderns have selected this vesti¬ 
 bule for the admission of the public to view the 
 museum of the exact sciences. 
 
 Prod us, the last Greek geometer, seems to have 
 had this idea. He alone, of the ancients, is recorded 
 to have paid distinguished attention to the First Book 
 of Euclid. He left a special commentary upon it, 
 abounding in curious observations on the metaphysics 
 of Euclid, and with curious facts connected with the 
 science. 
 
 The desirability of recasting our educational trea- 
 
MATHEMATICS. 
 
 11 
 
 tises the commercial habits of the age leave be¬ 
 yond doubt. The industrious classes are they who 
 most need the acquisition of any logical power which 
 Geometry can impart, and their means of leisure to 
 pursue such a study are few and far between. Ac¬ 
 tuated, probably, by convictions of this kind, Sir John 
 Herschel has composed a work on Astronomy, in¬ 
 tended to answer for the astronomical, such purposes as 
 are here contemplated for the geometrical public. 
 Indeed, many distinguished men have turned their 
 attention to the same object in their respective sciences. 
 Of the fitness of the single book of Euclid for the 
 accomplishment of the ends here specially in view, 
 the reader will be able to judge from the exe¬ 
 cution of these pages. Since the -writer first arranged 
 them, a coincident opinion has been very distinctly 
 expressed upon the subject.— 
 
 ‘ To young readers, about to enter on the study of 
 Astronomy, and who seek only to get such a mere # 
 general knowledge of it as may satisfy their own 
 minds, we would say in the first place, acquire a 
 knowledge, however slight it may be, of the elements 
 of Mathematics. Your mind may not be of a mathe¬ 
 matical turn, and there may not be the slightest pros¬ 
 pect of your deriving any positive advantage from 
 posing your brains with the “ First Book of Euclid." 
 No matter; try and go over it; it is worth your 
 while. You cannot stir in Astronomy without know¬ 
 ing something of the properties of the circle and tri¬ 
 angle. He, therefore, who wishes to comprehend the 
 “ reasons ” on which Astronomy is based, will acquire 
 this preliminary knowledge, without which it is useless 
 for him to enter upon the study. After he has ac¬ 
 quired it, and after he has studied an Astronomical 
 work, he may be far—very far, indeed, from having 
 the smallest pretension to the name of Astronomer, 
 
12 
 
 INTRODUCTION, liTC. 
 
 }»llt lip will Ka in __ !_ P n n 
 
 acquaint themselves -with the entire productions of 
 these authors, have become familiar, through such con- 
 vcnient epitomes, with their merits and their graces. 
 That there are both romance and beauty of a stem kind 
 in the wonderful writings of Euclid, no man of infer- 
 mation or taste will deny. His works were the torch 
 w hich lighted the sciences to their earliest discoveries. 
 They revealed to us the world of form and the 
 hidden secrets of figure, and enlisted lines and 
 angles in the eternal service of man. The deep en¬ 
 thusiasm which some have felt for Geometry will 
 he generally shared in the current diffusion of 
 intelligence. An important connection is beginning 
 to be established between the principles of science and 
 morality. More than one philosopher is found to ac¬ 
 knowledge, that ‘The axioms of Physics translate 
 those of Ethics.’f The desire of aiding to some extent 
 the development of this fact was not one of the least 
 reasons which has dictated the compilation of the 
 ‘ Beauties and Uses of Euclid.* 
 
 * London Saturday Journal, 
 f Emerson. 
 
CHAPTER II. 
 
 IMPORTANT DISTINCTIONS PERTAINING TO 
 MATHEMATICS. 
 
 Mathematics is the science of quantity and magni¬ 
 tude. It is usually defined as that which treats of // 
 whatever can be numbered or measured.* Mathe- / 
 matics embraces all modes of procedure, and includes 
 all the arts and sciences employed in determining 
 magnitude and quantity. Quantity is understood to 
 include the distinctions of weight and measure. Mag¬ 
 nitude is the consideration of size. 
 
 Geometry , though a very important science, is but 
 a branch of the Mathematics. Geometry treats of 
 the properties of lines, angles, surfaces and solids. 
 
 The term Geometry is applied to Mensuration, to 
 Trigonometry, Conic Sections, and to every depart¬ 
 ment of investigation in which lines and angles, sur¬ 
 faces and solids, are chiefly concerned. 
 
 Arithmetic, the art of figures — and Algebra, 
 the science of symbols, constituting the lower and 
 higher departments of calculation, are both branches 
 of Mathematics. But Geometry is the principal 
 branch. 
 
 From these points of definition the relative dis¬ 
 tinctions subsisting among the branches of Mathe¬ 
 matics, will easily be perceived, and the careful obser¬ 
 ver will be enabled to avoid the error of confounding 
 these distinctions. The common mystery attached to 
 the Mathematics has, in a great measure, its rise in 
 this kind of inattention. 
 
 ♦Ritchie’s Principles of Geometry, page 2. 
 
14 
 
 INTRODUCTION TO TIIE 
 
 A person skilled in the use of figures is an ArithnJM 
 tician. This is hispropcr appellation. Ifacquuiated with 
 that higher branch of calculation in which quantities 1 
 are represented by letters, his correct designation is an 
 Algebraist. The proficient in Euclid’s Elements is a 
 Geometrician. The more precise ancients always 
 styled the illustrious author of the Elements the 
 Great Geometer. It is the person who has ac¬ 
 quired skill in all these branches, and such others as 
 pertain to the determination of magnitude and quan¬ 
 tity, who is properly the Mathematician. As Mathe¬ 
 matics includes all these arts, properly, the Mathema¬ 
 tician is master of them all. 
 
 We often, indeed, hear the skilful Algebraist styled 
 a Mathematician. The Trigonometrist and Geome¬ 
 trician often receive the same designation. 13 ut 
 correct taste is never guilty of this confusion of 
 terms. Such want of precision is a poor compliment 
 to the precisest of sciences. 
 
 The term Mathematics will never be intentionally 
 employed in this work, except in its comprehensive 
 sense, as explained in its definition. That term on 
 the title page is employed to indicate that an attempt 
 will be made to lay bare its clear import, and by so 
 doing divest it of its reputed mystery; and illustrat¬ 
 ing (as will be done in various places) the exact na¬ 
 ture and influence of its branches, and afford a com¬ 
 prehensive and easy insight into its nature and applica¬ 
 tion. But chiefly, this work, as its second title ex¬ 
 presses, is confined to Euclid—to Geometry. 
 
 The Elements of Euclid arc often designated the 
 Elements of Mathematics. True, the Elements once 
 constituted Mathematics themselves; but since the 
 perfection of numerals and the invention of Algebra 
 by Diophantus, they have sunk into a relative rank. 
 It is true, Geometry gives laws to Arithmetic. The 
 
MATHEMATICS. 
 
 / 
 
 dual rise out of the wants of men by patient study, 
 and its progress, will further divest him of the preju¬ 
 dice, if any remain, of the popular mystery connected 
 with this science. 
 
 CHAPTER III. 
 
 HISTORY OF THE RISE AND PROGRESS OF THE 
 MATHEMATICS.* 
 
 ‘The Mathematical Sciences were the first of all 
 among men, if we may believe Josephus. He (Bk. I. 
 Chap. 3,) writeth, that the posterity of Seth observed 
 the order of the Heavens, and the courses of the Stars. 
 And lest these inventions should slip out of the know¬ 
 ledge of men, Adam having predicted a two-fold des¬ 
 truction of the Earth, one by a Deluge, and the other 
 by Fire, they raised two Columns, one of Bricks, the 
 other of Stone, and inscribed their inventions upon 
 them ; and if the Brick one should happen to be 
 destroyed by the Deluge, that of Stone, which would 
 remain, might afford men an opportunity of being 
 instructed, and present to their view the things which 
 had been inscribed upon it. They also say, that that 
 
 * Compiled chiefly from Andrew Tacquet’s (of the ‘ Society 
 of Jesus’) Elements of Euclid, translated I»y Whiston—the 
 Introduction to the Elements of Geometry, translated into 
 English from Peyzard, by Phillips—and from the History of 
 the Science prefixed to the Elements of Plano Geometry, by 
 Bell. But, of course, omitting the usually recited lists of the 
 inventions of the ‘ Fathers of the Science.’ If enumerated, 
 who, on a general introduction to Mathematics, would feel 
 enraptured or be enlightened by details of ‘ The Lair,’ ‘ The 
 Loci,’ Pori, Quadratics, and Curves ? They arc best intro¬ 
 duced where they are intelligible. 
 
1G 
 
 INTRODUCTION' TO THE 
 
 Rts arc h ~ nuB and o^glc *. It < io' ft i liutint l , e ntire, 
 
 and independer t science 
 But it will fbe asked 
 Mathematics 
 Differential a 
 but also sue 
 measurement 
 Geography, 
 
 by the/intelligent trader, if 
 omprises not only/Arithmetic,/Algebra, 
 d Integral Calculus, and Geometry— 
 principles as are employed in the 
 f the earth and taie skies, in Physical 
 'Astronomy, Navigation, Aifchitecture, 
 and a varietV of other branches, how is the maxim 
 with which pec. I. opens to ha applied by the general 
 enquirer to these departments? The answer is easy. 
 
 If the enfluirer aims at the eminent distinction of 
 being a Mathematician, he must prosecute each of 
 these brancmes of study, so "as to acquire familiarity 
 with their modes of procedure, and somq skill in their 
 application. Nor will this be so difficult as at first 
 sight appears. The nature'of each is {kindred, and 
 Geometry is the key which j( ill unlock them all—only 
 it requires the application or many years to compass 
 so wipe a range of physical knowledge. 
 
 course of the geiteral enquirdr is different. 
 Arithmetic being requiretj, to some extent, by every- 
 
 ti 
 
 Dther arts. 
 
 for Arithmetic and several other arts. UTio 
 and IP radical Geometer, (^'distinction hereafter to 
 plainer) come to Euclid for .the reasons of t 
 aware that Schiller (according to the Dublin 
 zine) mpjees tho great distinction between art 
 
 Arithmetician 
 be ex- 
 ir rules. I am 
 diversity Maga- 
 d pure science 
 
 to consikt in the element /of humanity being always essen¬ 
 tially involved in every work of art, and this element being 
 excluded in all that is properly called science, llut this dis¬ 
 tinction mates to high art—to the poetry of Wt. I have 
 here to do with art in the * sense of manipulatioiA-ahd'Vfth • 
 science in the sense of systematized and demonstrative know¬ 
 ledge—and m every employment of these terms, Vllusion is 
 made to the\irt and sdlcnco of practice. No disparage¬ 
 ment to Schiller's distinction, but ours is the othei depart¬ 
 ment. f 
 
MATHEMATICS. 
 
 17 
 
 body* he acquires .its first principles, and as much or 
 its praqtice as' lfe'is likely td"!Te6d‘.' ""If' he is curious 
 about Algebra, he may soon gather ' from Coles’s 
 admirable ‘ Conversfmons’ on that subject, the requisite 
 information—the Fi|s* Book of Euclid will accomplish 
 the rest. It will introduce him to both Theoretical 
 and Practical Geometry, and to the technical charac¬ 
 teristics of Mathematics. It will enable him to under¬ 
 stand the modus operandi of Newton’s Principia. It 
 will familiarise him with the illustrations of Dynamics 
 and Mechanics. After performing the * Constructions 
 of Euclid, and putting them to the * Uses ’ at the end 
 of this book, any personjcould understand and practice 
 Keith’s principles of Mapping, and illustrate his 
 geography by constructing his own maps. We have 
 seen in the quotation in the last chapter, from the 
 Saturday Journal , how tthe First Book of Euclid, if 
 it does not make the young pupil master of Astronomy, 
 it will place him on the lame common ground with 
 the Astronomer himself, arid familiarise him with the 
 scientific secrets of that fine pursuit. Indeed, if the 
 First Book of Euclid do&s not make the student a 
 Mathematician, it so qualifies him that he can be a 
 Mathematician whenever He pleases, as it imparts a 
 clear insight of that science which will enable him to 
 extend his studies whenever his duty or taste shall 
 incline him. Thus the general enquirer applies our 
 maxim. Fie determines wllat is properly worth his 
 attention, and earnestly devotes himself to that, and 
 to nothing else. His business is to explore , not to 
 conquer the region of Mathematics It is that which 
 
 i t io iworth his uhile t xi tfo, audpuF tIt^ ffrff r'w r; l h -- 
 
 Mirn y people of undisciplined powers, and ignorant 
 of the art of acquiring knowledge, can never move 
 anywhere unless they move profoundly. Such people 
 soon lose themselves. Instead cu being the masters of 
 
18 
 
 INTRODUCTION TO THE 
 
 knowledge, knowledge'is the master of them u. 
 
 -ut h^ a 4; 0 z^zWE S 4 ? 
 
 One subject only will one genius fit. 
 
 So vast is art—so narrow human wit. 
 
 in^ n0 £ be deter , red from the useful course of general 
 £?Zn m th i se two Sections recommenced bv 
 the v % U o charge ^superficiality, so often preSred 
 
 thout propriety and listened to without advantage 
 
 EXTS? IVl 1 * e “ *££ 
 
 SupeXiS ia » d * erm r d »ot to be superficial, 
 cupemciahty 18 a serious imputation when it respects 
 what you ought to know well. But with reS to 
 gCT^ral knowledge, it is no more criminal to be super- 
 t0 have onI y five senses, or not to be in half 
 
 only to fifS at + ° n fi 11 iS thG C ° ndition of ^manity 
 fiS lfl 6 i t0 ^e a general survey of the wide 
 
 inant ^ n °b edge ‘ + ^ re P ublic of literature has no 
 vhfeh ™ CQ t0 th6 1 6tate , ° f S ood citizenship, in 
 
 Z 1““ 18 ? P<3Cted t0 know his own business, 
 SUC1 , a n acquaintance only with that of 
 his neighbours as shall give him a laudable interest in 
 
 USSUrfe him tha t they are conducive 
 to tup dignity of the commonwealth. 
 
 t!ui UlUluiily uf tlitf brnnnliou nt 
 
 master the first 
 
 the 
 
 does_ 
 
 ^work that bun 
 
 we uur: 
 
 principles. 
 
 "tfit has ever 
 in so striking 
 .of Euclid. Byt 
 now in- 
 
 hich ho is prepared. The record: ‘ wa^ ^ 1CS, 
 
 only 
 
 happens _ . 
 
 devised teaches 
 a manner as 
 more of 
 trodui 
 
 First 
 
 proper place. 
 
 'eader to the Histor 
 
0 
 
 mathematics. 19 
 
 dual risk out of the wants of men by patient study, 
 and its Progress, will further divest him of the preju¬ 
 dice, if taiy remain, of the popular mystery connected 
 with thislscience. 
 
 CHAPTER III. 
 
 IIJSTOHY OF THE m fm-A*.™ mnniH^w rtTTf 
 MATHEMATICS.* 
 
 ‘The Mathematical Sciences were the first of all 
 among men, if we may believe Josephus. He (Bk. I. 
 Chap. 3,) writeth, that the posterity of Seth observed 
 the order of the Heavens, and the courses of the Stars. 
 And lest these inventions should slip out of the know¬ 
 ledge of men, Adam having predicted a two-fold des¬ 
 truction of the Earth, one by a Deluge, and the other 
 by Fire, they raised two Columns, one of Bricks, the 
 other of Stone, and inscribed their inventions upon 
 them j and if the Brick one should happen to be 
 destroyed by the Deluge, that of Stone, which would 
 remain, might afford men an opportunity of being 
 instructed, and present to their view the things which 
 had been inscribed upon it. They also say, that that 
 
 • Compiled chiefly from Andrew Tacquet’s (of the ‘Society 
 of Jesus’) Elements of Euclid, translated by Whiston—the 
 Introduction to the Elements of Geometry, translated into 
 English from Peyzard, by Phillips—and from the History of 
 
 Bell. .But, 
 
 inventions 
 
 ^i-cowseynniHt'hig'thu usually Ter 
 
 ItTil 1 i."! i 1 ) "nf the 
 
 who, on a 
 enraptured 
 Loci,’ Pori 
 duced wlier< 
 
 general introduction to Mathemat 
 or be enlightened by details of ‘ '1 
 Quadratics, and Curves ? They 
 they are intelligible. 
 
 cs, would feel 
 he Lair,’ ‘ Tho 
 are best intro- 
 
INTRODUCTION TO THE 
 
 20 
 
 Stone Pillar, which even in our day is seen in Syria 
 was dedicated by them. This Josephus says, whom 
 1 leave to vouch for the story.’* 
 
 Geometry must be coeval in its origin with the in¬ 
 vention of the square and the compasses, and these 
 being necessary to the rudest operations, were doubt¬ 
 less known in the earliest ages of the world. Geometry 
 first assumed the character of a science, in Greece, 
 about 2,500 years ago, and flourished during ten 
 centuries: then was almost entirely neglected till 
 about 200 years since, when it broke forth in Europe 
 with more than its pristine brilliancy. 
 
 The term Geometry is derived from the Greek 
 words ge (earth) and matron (measurement), literally 
 signifying the Measurement of the Earth. But the 
 science thus designated is now so extended, as to be 
 equally applicable to the measurement of the heavens. 
 Aristotle, the famous preceptor of Alexander the 
 Great, ascribes its origin to the Egyptian Priests, 
 and Herodotus, an ancient historian, to the time when 
 Sesostris, who reigned in Egypt, (1500 B. C.)+ inter¬ 
 sected it by numerous canals, and apportioned the 
 country among the inhabitants. 
 
 To Egypt appears chiefly due the honour of having 
 been, if not the birth place, at least the cradle of Geo¬ 
 metry, where it received its first culture. That old 
 and mighty kingdom seems to have been the nurse, 
 perhaps the mother, of almost all the useful arts and 
 sciences. Hindoostan and China contend with Egypt 
 for the pre-eminence with respect to Geometry; but 
 by whichever it may have been first cultivated, it 
 
 * Tacquet’s Historical Account of the Rise and Progress 
 of Mathematics. 
 
 + B. C. Before the reputed Birth of Christ—before the 
 commencement of what is denominated the Christian Era. 
 
MATHEMATICS. 
 
 21 
 
 was certainly brought into general notice by the Egyp¬ 
 tians. 
 
 Thales, the reputed father of Greek philosophy, born 
 at Miletum (640 B. C.), is celebrated for being 
 the first who introduced Geometry to Greece, and 
 for the many highly interesting discoveries he made 
 in it: for being able, by geometrical principles, to 
 compute the distance of vessels from the shore, and 
 for Having measured* the altitude of the Pyramids of 
 Egypt by means of their shadow, much to the as¬ 
 tonishment of Amasis the king, who wondered at his 
 scientific skill. From him Astronomy is said to have 
 made considerable advances—and he founded an estab¬ 
 lishment in Greece known by the name of the Ionian 
 School. It was in the age of this Philosopher, accord¬ 
 ing to the best informants, that Geometry first began 
 to assume the character of a science. He observed 
 the Equinoxes and Solstices, and was the first who 
 foretold the Eclipse of the Sun. And he foretold to 
 King Cyrus an Eclipse of the Moon. The Electrical 
 properties of amber were first remarked by him, upon 
 which has been built the important science of Elec¬ 
 tricity. 
 
 The first elementary treatise on Geometry is said 
 to have been composed by Anaximander, a disciple ot 
 Thales, and Thales’s successor in his school. 
 
 Next in order comes Pythagoras,f the philosopher 
 
 * Thales, one of the seven wise men of Greece, about COO 
 years before Christ, invented the following method for mea¬ 
 suring the height of an Egyptian pyramid. He watched tho 
 progress of the sun till his body and shadow were of the same 
 length, and at that instant measured the shadow of the 
 pyramid, which consequently gave its height.— Lord Kame's 
 History of Man, vol. 1, p 91. 
 
 t The supposed inventor of the term ( Philosophy ’— sig¬ 
 nifying a lover of knowledge. A term said to have been first 
 applied to Solon, an ancient law-giver. 
 
22 
 
 INTRODUCTION TO THE 
 
 of Samos, (an island in the Levant, between Greece 
 and Asia), no less distinguished than Thales for the 
 variety and extent of his discoveries. He is said to 
 have travelled into Egypt and India in pursuit of 
 knowledge, to have settled at Tarentum in Italy, 
 (about 550 years B.C.), and to have maintained, in 
 Astronomy, the true system of the world which places 
 the sun in the centre with the planets revolving 
 round it. From this philosopher it was called the 
 Pythagorean System, and was revived two thousand 
 years after his day by Copernicus. Pythagoras was 
 distinguished in all his pursuits by a genius remark¬ 
 ably inventive and assiduous. He possesses no ordi¬ 
 nary claim to the honour of posterity. As a mathe¬ 
 matician, he was decidedly the first of his time,—as a 
 philosopher, he delivered many curious things con¬ 
 cerning God and the human soul, and a great variety 
 of precepts relating to the conduct of life, both civil 
 and political. 
 
 It is not certain at this time whether Geometry had 
 been founded into a regular system. If not, consider¬ 
 able advances had been made, and ere a century had 
 elapsed from the age of Pythagoras, Zendorus, a man 
 of great parts, arose, whose writings are the first 
 among the ancients which have survived the wreck of 
 time, one geometrical tract of his having been pre¬ 
 served. 
 
 Then came Hippocrates, whose brilliant genius, 
 great diligence, and application, rendered essential 
 service to this science. He was the third who wrote 
 on the Elements of Geometry, but on the appearance 
 of the Elements of Euclid, his treatise was consigned 
 to oblivion. 
 
 The origin of the celebrated Platonic School, (400 
 years B.C.), is considered one of the most important 
 epochs in the history of the science. Plato, its 
 
MATHEMATICS. 
 
 23 
 
 ? founder, who is called by way of pre-eminence the 
 < Athenian Sage/ and who was alike remarkable for 
 eloquence, the poetical sublimity of his imagination, 
 and the accuracy of his mathematical performances, 
 gave to Geometry the form and substance of a com¬ 
 plete science, and enriched it with many discoveries. 
 Indeed, so profound a veneration did he entertain for 
 it, that he made it a principal object of instruction 
 among his scholars, and had written over the entrance 
 of his academy, the well known words ,—‘ Let no one 
 enter here who is ignorant of Geometry .’ Plato was 
 the discoverer of the Conic Sections, (or those curves 
 which are found on the surface of a Cone when cut 
 through in different directions). He found out many 
 of their most remarkable properties, which being 
 made the continual study of his scholars and succes¬ 
 sors, at length became a distinct science from the 
 common elements, and received the appellation of the 
 higher or sublime Geometry. 
 
 Such was the versatility of his genius, that he 
 directed his acute and original powers with eminent 
 success to the cultivation of Moral Philosophy, and 
 produced his famous work, entitled the * Republic/ 
 with the view of imparting to civilised intercourse 
 the character of a science, and giving a philosophical 
 direction to the ever jarring aims and interests of 
 mankind. 
 
 The celebrated Aristotle, the successor of Plato, 
 an<fr preceptor to Alexander the Great, and who is 
 justly regarded as one of the greatest men of olden 
 time, made many improvements in the mathematical 
 sciences. After attending the lectures of Plato, he 
 opened a school himself in the Lyceum of Macedonia, 
 which was assigned him by the magistrates, and -was 
 the founder (about 350 years B.C.), of the well known 
 Peripatetic school. 
 
24 
 
 INTRODUCTION TO THE 
 
 Theophrastus, a disciple of Aristotle, composed a *3 
 complete history of the origin and progress of Mathe¬ 
 matics, Astronomy, and Arithmetic, from the earliest 
 periods to his own time. The treatise has been unfor¬ 
 tunately lost. 
 
 Archytas, of the academy of Plato and of Taren- 
 tum, the place in which Pythagoras settled, is famous 
 as the constructor of a flying pigeon, and as the re¬ 
 puted inventor of the pulley and screw. ‘ He was one 
 of the first,’ says Tacquet, 1 who brought down the 
 Mathematics to human uses.’ Aristoeus, who flour¬ 
 ished (about 300 years B.C.), wrote many works of 
 considerable merit, acquired great proficiency in 
 Sublime Geometry, and is said to have been the in¬ 
 structor of Euclid. 
 
 Euclid, according to Pappus and Proclus, was born at 
 Alexandria,* in Egypt, where he flourished and taught 
 Mathematics with great applause, under the reign of 
 Ptolemy Lagus, (about 280 years B.C.) Some Ara¬ 
 bian historians, however, inform us, that he was born 
 at Tyre, that his father’s name was Nancrates, an in¬ 
 habitant of Damas. The particular place of his na¬ 
 tivity appears, therefore, uncertain, but whether or 
 not Alexandria had the honour of giving him birth, 
 all historians agree that he flourished there as a 
 teacher at the time above mentioned—which city, from 
 that period till the time of its conquest by the Saracens, 
 seems to have been the residence, if not the birth ^ce, 
 of all the most eminent mathematicians. It is sup¬ 
 posed that Euclid studied at one time under the disci¬ 
 ples of Plato at Athens. History is silent as to the 
 time of his death. Pappus represents him as a per¬ 
 son of great courtesy, mildness, modesty, and benevo- 
 
 * A town built by Alexander the Great, near tlic mouth of 
 the Nile, in Egypt. 
 
MATHEMATICS. 
 
 23 
 
 knee: lie left various profound treatises on the most 
 abstruse subjects belonging to his profession, but 
 his eminent work is his Elements of Geometry, 
 which supplanted all of a similar kind that preceded 
 it and such was the judgment he displayed in its 
 composition, that notwithstanding the great additions 
 made to Geometry since his time, it has continued for 
 upwards of 2000 years to sustain the highest reputa¬ 
 tion as an elementary treatise.* 
 
 The great reputation of Euclid arose from the ex¬ 
 cellence of his Elements. His merit in this perform¬ 
 ance consisted in his having systematised the scattered 
 discoveries of Thales, Pythagoras, Plato, and the 
 lesser geometers who preceded him. No doubt he 
 supplied important links. But when he infused co¬ 
 herence into the fugitive inventions of his predeces¬ 
 sors, he imparted to Geometry the importance of a 
 science. 
 
 Next after Euclid came Archimedes, who was born 
 in the Island of Sicily. He established the true 
 principles of mechanics, laid the foundation of Hydro¬ 
 statics,! traced out the first rudiments of Naval 
 Architecture, and applied his talents with such 
 success to the cultivation of Geometry, Arithmetic, 
 and Optics, that he has been styled the Newton of 
 antiquity. He is said to have prosecuted his studies 
 to the neglect of both food and sleep. 1 He perished 
 
 * The curious may be interested to learn, that tho 
 Elements of Euclid were first published in Latin, in folio, by 
 Radholt, at Venice, in 1482. First published in the original 
 Greek by Hervagius, folio, Basil, 1533. And the first English 
 translation was by Billingsley, folio, London, 1570, with a 
 curious preface by John Dee. 
 
 t Tho scienco of weighing fluids, 
 t No great proof this of his ‘philosophy.’ Nothing is 
 philosophical which cannot with prudence be imitated. 
 
26 
 
 INTRODUCTION TO THE 
 
 at the fall of Syracuse. A soldier came suddenly 
 upon him in the museum, when he was intently en¬ 
 gaged on some geometrical problem, and commanded 
 him to follow him to Marcellus, (the Roman Consul 
 who had taken the place). He refused, however, to stir 
 till he had finished the subject on which he was en¬ 
 gaged, and the soldier ran Archimedes through with 
 the sword. When dying he desired that a sphere in¬ 
 scribed in a cylinder might be engraved on his tomb 
 to perpetuate the memory of his most brilliant dis¬ 
 covery.* The Sicilians, however, soon forgot him, 
 and his tomb was found 200 years after by Cicero, 
 by means of the symbol just named, in a field near 
 Syracuse, overgrown with thorns. He was the most 
 inventive, and the greatest Mechanical Engineer of 
 ancient time. 
 
 Eratosthenes, a native of Cyrene, was called to the 
 Alexandrian school by Ptolemy Euergetes, who made 
 him his librarian on account of his extensive and 
 general attainments in learning. His knowledge of 
 Geography, Astronomy, and Geometry, enabled him 
 to discover, for the first time on record, a method of 
 measuring the circumference of the earth. 
 
 Fifty years after the death of Archimedes, Apollo¬ 
 nius appeared, who gave a great impulse to the ma¬ 
 thematical sciences, and whose discoveries were so 
 Highly esteemed that he was honoured by the appel¬ 
 lation of the Great Geometer. 
 
 From the time of this eminent man we move on 
 for three or four hundred years without meeting with 
 one person who contributed to the advancement of the 
 sciences. 
 
 * He discovered some very fine proportions which exist be¬ 
 tween the sphere and the cylinder. 
 
MATHEMATICS. 
 
 27 
 
 Then came Theon, (about 380 years, A.D.*) who, 
 by his skill and perseverance in Mathematics and 
 Philosophy, obtained the honourable dignity of being 
 appointed president of the famous Alexandrian School, 
 where, by his condition and conduct, he gained the 
 greatest respect and reputation. 
 
 Theon had a remarkable pupil in his own daughter, 
 Hypatia—a lady learned as well as beautiful. She 
 was born at Alexandria. Hypatia was mistress of all 
 the ordinary accomplishments of her sex, familiar 
 with the most abstruse sciences, and made such 
 progress in Philosophy, Geometry, Astronomy, and 
 the Mathematics generally, that she was held as the 
 most learned person of her time. She published com¬ 
 mentaries on Apollonius’s Conics, on Diophantus’s 
 Arithmetic,! and other works. She was chosen whilst 
 very young to succeed her father in the same school, 
 and to discourse from the presidential chair, at a time 
 when Ammonius, Hierocles, and many other very 
 learned men abounded at Alexandria and in various 
 parts of the Roman Empire. Among her pupils, not 
 less eminent than numerous, was the much esteemed 
 Synesius, afterwards Bishop of Ptolemais. But it was 
 not Synesius only and the disciples of the Alexandrian 
 School who admired Hypatia for her virtues and learn¬ 
 ing. She -was held as an oracle by the public, and 
 was consulted by the magistrates on all important 
 occasions. In short, when Nicephorus intended to 
 pass the highest compliment on Eudocia, he thought 
 lie could not do it better than by calling her 
 
 * (380 A.D.)—380 years after Christ: A.D. being the ab¬ 
 breviation of Anno Domini, the year of our Lord. 
 
 f ' Diophantus was as great in Arithmetic as Archimedes, 
 Apollonius, or Euclid was in Geometry, and by him was found 
 out that admirable art we call Algebra.’— Tacquet. 
 
 C 
 
28 
 
 INTRODUCTION, I'.TC. 
 
 another Hypatia. Whilst Hypatia thus reigned 
 the most brilliant ornament of her sex in the annals 
 of history, she was greatly admired by Orestes, 
 the governor of the city, who, on account of her 
 wisdom, often consulted her. This, together with an 
 aversion which St. Cyril had against Orestes, proved 
 the cause of her ruin. About 500 monks assembled, 
 attacked the governor one day, and would have killed 
 him had he not been rescued by the townsmen—and 
 the respect which Orestes had for Hypatia, causing 
 her to be traduced among the Christian multitude, they 
 dragged her from her chair, tore her in pieces, and 
 burned her limbs.* This shocking and brutal catas¬ 
 trophe was perpetrated in the lent of the year 41G, in 
 the time of Theodosius II.f 
 
 Pappus, a consummate mathematician, flourished 
 about the end of the 4th century. His work, entitled 
 1 Mathematical Collections,’ has transmitted his name 
 to posterity with distinguished lustre. 
 
 Proclus, who was at the head of the Platonic 
 School, established at Athens (A.D. 500,) rendered 
 important service to the sciences, and showed great 
 kindness to all who embraced their pursuit. 
 
 The age of important discovery was past before 
 the time of Pappus—and science in his time was 
 beginning to decline. The labours of Proclus and a 
 few of his followers were the last expiring efforts 
 
 * It is related by Damasius and Suidas, and clearly proved 
 by such learned men as Bruker, La Croze, and Basnaoe, that 
 St. Cyril incensed the Christian populace against her.—( Vol- 
 Zaire's Phil. Diet. art. Ilypatia.) St. Cyril was a man of parts, 
 and there is a difficulty in accounting for the inhuman prin¬ 
 ciples that actuated him in this affair. 
 
 t For a more particular account of this illustrious victim 
 of fanaticism, see Bossut’s Hist. Mat., English Ed., Ovo., 1803. 
 
MATHEMATICS. 
 
 29 
 
 1 made in Greece for ancient Geometry. The Alexan- 
 I drian School still existed, and science might have con- 
 | tinued to flourish, had not the conquest of Egypt bv 
 the Arabians, who, with the design of spreading their 
 f religion and their empire, inundated Christendom in 
 the seventh century, and not only gave it a fatal blow 
 | in Alexandria, but also extinguished its now languid 
 L existence in Greece. 
 
 f- Throughout all the civilised parts of the world in 
 ' those times, the Alexandrian Library was known as 
 the treasure-house of literature and knowledge. Bet 
 f neither the members nor the materials of this sacred 
 edifice were respected by the Arabians, who, stimu¬ 
 lated perhaps more by religious fanaticism than bar- 
 ; barous ignorance, sacked Alexandria, and massacred 
 the philosophers who were assembled there for the 
 purposes of study from all the neighbouring countries. 
 Their laboratories and instruments were destroyed, and 
 ' finally the Library itself, the accumulated scientific trea¬ 
 sure of ages, containing above 200,000 written volumes, 
 was committed to the flames for the mean purpose of 
 \ heating the public baths. Caliph Omar, as he gave 
 t orders for the annihilation of the books, justified it by 
 the strange, but convenient argument, that—‘ if the 
 volumes were conformable with the Koran, they were 
 superfluous; and if contrary to it, impious.’ With this 
 brutal act of bigotry and devastation may be said to 
 have closed the scientific career of antiquity. 
 
 As a partial amends for so odious an outrage against 
 civilization, the Arabians, about a century after, began 
 to cherish the sciences of Geometry and Astronomy, 
 and thus afforded an asylum which preserved them 
 from entire extinction. About the time of their 
 learned prince Almamon, several of their most cele¬ 
 brated works on Geometry were translated into the 
 Arabic. Geber Ben Aphla laid the foundation of the 
 
30 
 
 INTRODUCTION TO THE 
 
 modern Trigonometry. An elegant treatise on Men¬ 
 suration was written by Bagdadin; and it appears 
 from his treatise on Optics, that Alharin was a superior 
 geometer. They studied also the more sublime Geo¬ 
 metry, improved Trigonometry, and reduced it to a 
 convenient form. 
 
 Of the few Persian geometers, Nassir-Edin and 
 Maimon-Reschid were the most distinguished. They 
 both composed commentaries on Euclid. Persian 
 learned men called Geometry the difficult science, 
 and did not make any improvements in it—neither 
 did the Turks nor Hebrews, although they translated 
 some of the elementary works. 
 
 In one of the ancient treatises on Astronomy of 
 the Hindoos, called Suryd Sidhanta, there is a correct 
 system of Trigonometry; and hence it is concluded 
 that the Indians had cultivated Geometry before the 
 composition of this treatise. The Chinese were ac¬ 
 quainted with the 47th of the First Book of Euclid. 
 
 The Romans knew little more of Geometry than 
 the practice of land measuring. Varro, however, 
 composed a treatise on Geometry, and Cicero had 
 some acquaintance and esteem for it too. Vi¬ 
 truvius and Boetius had some knowledge of Geo¬ 
 metry, but the latter is better known for nis ‘ Conso¬ 
 lations of Philosophy .’ 
 
 The learned Bede, who lived at the beginning of 
 the 8th century, was acquainted with all the branches 
 of the Mathematics. Under him Alcuin studied, who 
 was well versed in mathematics, and was pr eceptor to 
 the celebrated Charlemagne. He was a native of Bri¬ 
 tain, where, ab 0 ut this time, more attention was given 
 to the study of this science than in any country in 
 Europe. 
 
 For nearly two centuries after this period, the 
 study of the science was extinct. No mathematician 
 
MATHEMATICS. 
 
 31 
 
 of any eminence appeared till the beginning of the loth 
 century, when, with the general revival of learning, 
 this science was destined to commence a new and 
 more splendid career. 
 
 The Italians, by their intercourse with Arabia, be¬ 
 came, about the end of the 12th century, the restorers 
 to Christendom of its long-forgotten arts and sciences. 
 Leonardus Pisanus, a rich merchant of Pisa, who had 
 made, in the course of his profession, several voyages 
 to the East, is said to have been the first who dis¬ 
 seminated amongst Europeans a knowledge and a love 
 of mathematical studies, after both had disappeared 
 for so many ages.* 
 
 About the beginning of the 13th century, Campanus, 
 of Navarre, translated Euclid, and several geometri¬ 
 cal works were translated from Greek and Arabic 
 MSS., by Maurolycus, of Messina, who was dis¬ 
 tinguished for some original and elegant works of his 
 own. Ramus, or La Ramee, an original metaphysical 
 writer, also composed several excellent works on 
 Geometry : and Willibrad Snellius published at the 
 age of seventeen an attempted restoration of an ancient 
 work of much value. 
 
 About the beginning of the 16th century the 
 Spanish and Portuguese geometers, John of Royas 
 and Nonnius or Numer, lived. The former was the 
 inventor of the projection of the sphere, and the latter 
 was the first to introduce the Arabic System of Algebra 
 into Europe. About the same time, Wright, a skilful 
 
 f eometrrcian, invented his chart—improperly called 
 lercator’s chart. At this time, Germany produced 
 the three mathematicians, Werner, Rheticus, and Pitis- 
 
 * A MS., dated 1202, of this celebrated Italian, has lately- 
 been discovered by Cossali, a canon of Parma. 
 
32 
 
 INTRODUCTION TO THE 
 
 cus. Werner made a German translation of Euclid. 
 The Italian geometer, Lucus Valerius, determined the 
 centre of gravity in various mathematical figures, 
 conoids and spheroids. 
 
 Kepler -was the first to conceive the theory of mag¬ 
 nitude being composed of indefinitely small elements 
 of the same kind; as, for instance, that a circle is 
 composed of the surfaces of an indefinite number of 
 very small triangles having the centre for their common 
 vertex, and their small bases on the circumference. 
 By means of this theory, though not very satisfactory 
 in regard to scientific rigour, he extended Geometry 
 considerably beyond the stage in which it was left by 
 Archimedes. Cavallerius improved on the theory of 
 Kepler, and effected a still further extension of Geo¬ 
 metry by conceiving lines to be composed of points, 
 surfaces of lines, and solids of surfaces. 
 
 Galileo, so eminent for his discoveries in physical 
 science, did little to improve Geometry. He invented 
 the Cycloid, called the Helm of Geometry on account 
 of the discussions that originated concerning its pro¬ 
 perties. Gregory St. Vincent and Andrew Tacquet 
 were two Flemish mathematicians, the former prose¬ 
 cuted the quadrature of the circle—the latter, the 
 quadrature of surfaces and conic sections, with con¬ 
 siderable success. He rendered essential service to 
 the study of the science by subjoining the Uses to the 
 propositions in an edition of Euclid’s Elements, he 
 published, after the manner of the celebrated French 
 Jesuit, Claude Francois Milliett Dechales. Blaise 
 Pascal, a Frenchman, (born 1623, died 1602) 
 was equally distinguished for his great literary and 
 scientific attainments, was the author of several valuable 
 researches respecting the properties of the Cycloid; 
 and at the age of sixteen, he composed a complete and 
 elegant treatise of conic sections. But his health was 
 
MATHEMATICS. 
 
 &3 
 
 so much impaired, that soon after he was compelled to 
 suspend his studies for four years ; and he ultimately 
 perished at the age of 39, after a lingering illness, in 
 Paris. 
 
 Descartes, another eminent Frenchman, (born 1590, 
 died 1050), had a profound knowledge both of mathe¬ 
 matical and metaphysical science, was one of the 
 most efficient agents in accomplishing that intellec¬ 
 tual revolution which was in progress during his time, 
 and was the inventor of a method which produced a 
 thorough and memorable improvement in Geometry. 
 Before his time the application of Algebra to Geo¬ 
 metry, in the ordinary sense of the term, was known, 
 as examples of it are to be found in the works of some 
 ancient geometers. But Descartes invented that more 
 general mode of applying it—that the science of Geo¬ 
 metry, instead of being confined within the narrow 
 limits imposed by the complexity and inadequacy of 
 the ancient methods, is now of indefinite extent. 
 
 Iliiygens, a native of Holland, (1G29—1705) was 
 a profound mathematician. He solved a problem of 
 such difficulty, that it had not been previously attempted. 
 
 The year 1G12, which ‘gathered’ Galileo ‘ to his 
 fathers',’ gave to the world Isaac Newton, the Archi¬ 
 medes of modern times, or, as he has sometimes been 
 styled, from the number and importance of his dis¬ 
 coveries, the Prince of Modern Mathematicians. The 
 fame of Sir Isaac rests chiefly upon the application he 
 made of his acquisitions, in explaining the laws 
 or order by which the mechanism of the universe 
 appeared to him to be regulated. His views with 
 regard to this matter have been styled, in honour of 
 him, the Newtonian System of Philosophy, and have 
 been received implicitly by the learned world till nearly 
 the present time. 
 
 The impetus given to mathematical pursuits by 
 
INTRODUCTION TO THE 
 
 34 
 
 the genius of Newton, has descended to our day, 
 to use a phrase of his own, ‘with increasing mo¬ 
 mentum.’ For in his train have followed the illus¬ 
 trious names of Barrow, Keel, Gregory, Bishop 
 Horsley (editor of an edition of Euclid’s Elements), 
 the Rev. Mr. Lawson, Robison, and many others. 
 Among the most celebrated, rank Leibnitz, Euler,* 
 the Bernouillis, Clairant, D’Alembert,f Thomas Simp¬ 
 son, Robert Simson, the well known restorer, and 
 Playfair, the improver, of Euclid—Young, Leslie, and 
 Madame Agnesi,{ and Baron Swedenborg.^ 
 
 It is not necessary to extend this enumeration. The 
 mathematicians and geometers of this generation are 
 known to the public. But it is above estimate how 
 the diffusion of knowledge, one moment dreaded as 
 mischievous, and in the next with the customary in¬ 
 consistency of folly, pronounced impracticable, will 
 increase the public taste, for this science, and create 
 a just and enduring appreciation of this useful order 
 of scholars. 
 
 ♦ Euler lost his sight by intense application, and like 
 Milton, closed his labours in blindness. 
 
 t One of the writers in the celebrated French * Encyclo¬ 
 paedia.’ 
 
 < * This lady deserves the most respectful mention. Her 
 Algebraical Works, in quarto, translated by Olson, have 
 been long before the public, and very deservedly esteemed.’— 
 Harris's ‘ Compendium of Algebra.’ 
 
 § Swedenborg’s * Principia,’ a work declared, only very 
 lately by one of the highest professors in Europe, ‘ not un¬ 
 worthy of being placed by the side of Newton’s.’— Douglas 
 J err old's Mag., vol. 2, page 91. 
 
MATHEMATICS 
 
 30 
 
 CHAPTER IV. 
 
 UTILITY OF MATHEMATICS AS A MEANS OF EXTENDING 
 OUR KNOWLEDGE OF THE PHYSICAL WORLD. 
 
 Plato denied the world can he 
 Governed without Geometry. 
 
 Butler 3 & Hudibras. 
 
 It is a very trite observation, that human knowledge 
 has been greatly extended by means of this science. 
 Besides the important properties of magnitudes, and 
 wonderful relations of abstract quantities which it has 
 made known, it has unfolded a very extensive range 
 of natural phenomena. It has investigated the prin¬ 
 ciples of theoretical mechanics, the laws of equilibrium 
 and motion of fluids, fixed and elaslic. Optics, elec¬ 
 tricity, and magnetism are its debtors. The theory of 
 acoustics, the propagation of light, and numerous 
 other branches of science, have received great im¬ 
 provements by its agency. Even its reputed conceits 
 have been susceptible of useful application. The 
 ancient doctrine of the Conic Sections, which for 2000 
 years w r as an object of mere curious speculation, be¬ 
 came, in the hands of Newton, a most efficient means 
 of unfolding the planetary motions. 
 
 ‘ Without the aid of rules derived from this science, 
 the navigator, relying on his compass as a guide, could 
 not with safety venture to any considerable distance * 
 on his element; intercourse with transmarine regions 
 would be impossible; and, consequently, our know¬ 
 ledge of the globe we inhabit would be very limited. 
 We should probably yet believe that its surface is an 
 extended plane,and that it is supported on pillars; or, 
 as was the opinion of some ancient philosophers, that 
 
INTRODUCTION TO T1IF. 
 
 3(3 
 
 its figure is cylindrical, like a drum. Without the aid 
 of this science, our knowledge of celestial bodies 
 would be still more imperfect, and the consequences of 
 our ignorance still more striking. We should still 
 believe that these objects are equally distant from us, 
 and, very probably, that they are distributed on the 
 surface of an extensive crystalline sphere, performing 
 a diurnal rotation about the earth, as the centre of the 
 universe. We should also believe that some celestial 
 phenomena, as eclipses and comets, arc certain signs 
 of a conflict of the elements of nature, or that they 
 are the portentous indications of the wrath of heaven, 
 while contemplating to inflict on superstitious mortals 
 some dire calamity, as war, pestilence, or famine. 
 
 1 How different from these unsatisfactory and inco¬ 
 herent conjectures is that great achievement of this 
 science—the clear and satisfactory exposition, on the 
 most incontrovertible principles, of the complex though 
 sublime and systematic mechanism of the heavens ; 
 by which the distances and magnitude of the sun and 
 planets have been measured, and also their weights, 
 and even that of their satellites, ascertained; and by 
 which the masses and distances of some of the stars, 
 or suns of other systems, though inconceivably remote, 
 even in comparison with the great extent of our own 
 system, will probably ere long be determined.’* 
 
 An abler pen has celebrated this theme. £ The 
 scientific principles that men employ to obtain the 
 foreknowledge of an eclipse, or of anything else 
 relating to the motion of the heavenly bodies, are 
 contained chiefly in that part of science which is called 
 Trigonometry, or the properties of a triangle, which, 
 when applied to the study of heavenly bodies, is called 
 Astronomy; when applied to direct the course of a 
 
 * Bell. 
 
MATHEMATICS. 
 
 37 
 
 ship on the ocean, it is called Navigation; when ap¬ 
 plied to the construction of figures drawn by rule and 
 compass, it is called Geometry; when applied to the 
 construction of plans of edifices, it is called Archi¬ 
 tecture ; when applied to the measurement of the 
 surface of any portion of the earth, it is called Land- 
 surveying. In fine, it is the soul of science. It is an 
 eternal truth. It contains the mathematical demon¬ 
 stration of which man speaks, and the extent of its 
 uses is unknown.’* 
 
 The author of the Geometrical Companion defends 
 the practical uses of Geometry with the ardour of an 
 enthusiast. His examples are happy and forcible. 
 
 ‘ A bee, which is an instinctive geometer, finds the 
 use of it in constructing its comb, so as to be the most 
 roomy and durable at the least expense of time and 
 labour. Will any one after this fact doubt the use of 
 Geometry to an intelligent creature ? 
 
 ‘Almost all the evolutions of an army consist in 
 converting one parallelogram into another. And the 
 works of a fortified town, or position, are little else 
 than combinations of parallelograms forming bastions, 
 curtains, and parapets.’! 
 
 In our ‘ Uses/ other applications of this science will 
 be seen. Perspective, Dialling, and a variety of the 
 arts, might further be adduced. Literature gathers 
 from it many of its prettiest similes. Dr. Johnson has 
 two examples in his Life of Addison :—‘ A simile may 
 be compared to two Ivies converging to a ‘point, and 
 it is more excellent as they approach from a greater 
 distance/ 
 
 ‘ An exemplification resembles two parallel lines, 
 which run on together without approximation, never 
 far separated, and yet never joined. 
 
 * Paine. 
 
 + Darley. 
 
38 
 
 INTRODUCTION TO TIIK 
 
 Emerson lias a fine thought adorned with the 
 vestments of geometry. ‘ Every natural process is 
 but a version of a moral sentence. The moral law 
 lies at the centre of nature, and radiates to the cir¬ 
 cumference.’ 
 
 CHAPTER V. 
 
 MATHEMATICS AS A MEANS OF MENTAL DISCIPLINE. 
 
 The same excellent persons in the commonwealth of 
 learning who discovered Philosophy discovered also the Ma¬ 
 thematics : like twin sisters they came into the world, and 
 like twin sisters there is a natural affinity between them.— 
 Tacquet. 
 
 It is a mortifying reflection that the soberest of 
 sciences should have been disfigured by the exaggera¬ 
 tions of untutored enthusiasm. The Mathematics 
 have been extolled as the only source of logical 
 acumen and cool judgment. Its discipline has been 
 assumed to be the most salutary to which the human 
 powers can be subjected. Some indiscreet votaries 
 have carried its principles out of their legitimate pro¬ 
 vince, and applied them to subjects they were never 
 fitted to investigate. As might be expected, these pro¬ 
 ceedings have led to the disappointment of those who 
 believed such unwarranted pretensions, and to the ulti¬ 
 mate undervaluing of the real and substantial merits of 
 Mathematics. No more than the varying constitutions 
 of mankind admit of one panacea lor human ailments, 
 do the variations in human intelligence admit the pre¬ 
 tensions of one science to train up the faculties of every 
 individual. Natural Philosophy, Political Economy, 
 Morals, Metaphysics, each has its peculiar discipline 
 by which it addresses peculiar persons, and matures 
 
MATHEMATICS. 
 
 39 
 
 their powers where Mathematics would entirely fail. 
 The discreet friends of Mathematics never hesitate to 
 allow this, and with the judicious discrimination of 
 Tacquet, care only to claim for it kindred affinity to 
 Philosophy, and the moderate but just honour of its 
 having shed a certain lustre over human manners, and 
 the arts and the sciences at large. 
 
 Dr. Arnott, author of the Elements of Physics, 
 has attacked the assumed pretensions of -Mathema¬ 
 tics, and though he is far from doing it skilfully, it 
 cannot be denied that he had occasion for his reflec- 
 tions. * There is/ he says, ‘ no occupation which so 
 much strengthens and quickens the judgment as the 
 study of Natural Philosophy. This praise has usually 
 been bestowed on Mathematics; yet a knowledge of 
 abstract Mathematics existed with all the absurdities 
 of the dark ayes; hut a familiarity with Natural Phi¬ 
 losophy, which comprehends Mathematics, and gives 
 tangible and pleasing illustrations of its abstract 
 truths, seems incompatible with any gross absurdity. 
 A man whose mental faculties have been sharpened by 
 acquaintance with these exact sciences, in their com¬ 
 bination, and who lias been engaged, therefore, in 
 contemplating real relations, is more likely to discover 
 truth in other questions, and can better defend himself 
 against sophistry of every kind. We cannot have 
 clearer evidence of this, than in the history of the 
 sciences, since the Baconian method of reasoning by 
 induction took the place of the visionary hypotheses of 
 preceding times.’ 
 
 Dr. Arnott does not seem an enemy to pretension, pro¬ 
 vided lie is allowed to advance it. The kind of emi¬ 
 nence lie denies to Mathematics, and which might safely 
 be denied of any science, he does not hesitate to claim 
 for Natural Philosophy. He unhesitatingly asserts 
 that 1 there is no occupation which so much strengthens 
 
 D 
 
40 
 
 INTRODUCTION TO THE 
 
 the judgment as the study of Natural Philosophy.* * 
 But common experience, in the variety of intellectual 
 aptitudes, teaches us the fallacy of this broad asser¬ 
 tion. It is not a disparagement, as Dr. Arnott sup¬ 
 poses, that ‘ Mathematics existed with all the absurdi¬ 
 ties of the dark ageson the contrary, it is no little 
 praise that Mathematics had vitality enough to exist at 
 all under that weight of superstition and barbarism, 
 which buried all other learning in undistinguished 
 ruin. 
 
 A geometrician, whose ingenious performances I 
 shall have several occasions to cite, and whose popular 
 pen will give him considerable influence in other 
 quarters, has contributed a eulogium to his favourite 
 science, which cannot be permitted to pass unnoticed. 
 
 ‘Geometry not only sharpens our faculties, corrects 
 precipitancy, and prevents liability to imposition, either 
 from the arguments of others or our own, but it gives 
 us a satisfaction in our knowledge [which we could 
 not otherwise obtain], and thereby enables us to speak 
 with a strength and perspicuity which a vague per¬ 
 suasion of the truth of what we utter could never 
 inspire.’ 
 
 Mr. Darley affirms too much. He plainly alludes to 
 ‘ satisfaction’ in knowledge in general, and the words I 
 have put in brackets should have been omitted, as men 
 can, for thousands do, ‘ otherwise’ succeed in securing 
 that satisfaction.* It is enough to contend, which 
 
 * It is but fair to this writer to state that he has deduced 
 a compliment to Geometry from the biography of distin¬ 
 guished men, expressed with perfect propriety:— 
 
 * Alexander, Cresar, Charles l'2th, Buonaparte, and two of 
 the most celebrated military engineers, Vauban and Coehorn, 
 were profound geometers. And it is to their knowledge of 
 this science, the superior excellence of their systems must be 
 attributed .’—Geometrical Compa?iion. 
 
MATHEMATICS. 
 
 41 
 
 would be true, that the study of Geometry is one of 
 the modes of educational discipline, and a most ap¬ 
 proved method indeed. It is contending that it is the 
 only method which discovers vanity and offends good 
 
 taste. 
 
 When a stranger, an enemy, or an indifferent person, 
 chooses to laud Mathematics to the disparagement of 
 other branches of knowledge, the encomium may be 
 tolerated, though by no means imitated. Dugald 
 Stewart, who had no particular predilection for the 
 Mathematics, thus speaks of its study in his Philoso¬ 
 phy of the Human Mind :— 
 
 ‘ The intellectual habits of the metaphysician afford 
 little or no exercise to that species of attention which 
 enables us to follow long processes of reasoning, and 
 to keep in view all the various steps of an investiga¬ 
 tion till we arrive at the conclusion. Such processes 
 are much longer in Mathematics than in any other 
 science; and hence the study of it is peculiarly calcu¬ 
 lated to strengthen the power of steady and concatenated 
 thinking ; a power which, in all the pursuits of life, 
 whether speculative or active, is one of the most valu¬ 
 able endowments we can possess.’ 
 
 The Geologist, the Astronomer, the Chemist, seve¬ 
 rally have lauded their respective sciences as the most 
 important that can engage the attention of mankind, 
 both in an absolute and educational sense. This is the 
 language of ignorance or vanity, Mathematics need 
 never descend to the disparagement of other sciences. 
 It has its distinctive merits: it can bear any com¬ 
 parison, its province is peculiar, its utility unques¬ 
 tioned, and its method of procedure instructive. 
 These facts are its proper eulogies, and time has 
 failed to obscure, or the envy of ages to diminish, 
 their lustre. 
 
 Some of the features of discipline which indisput- 
 
42 
 
 INTRODUCTION TO THE 
 
 ably characterise mathematical study, have been well 
 depicted by a modem writer. 
 
 ‘ In reasoning, as in other arts, we are not masters of 
 what we do, till we do it both well and unconsciously: 
 now this advantage a judicious cultivation of Ma¬ 
 thematics supplies. It familiarises the student with 
 the usual forms of inference, till they find a ready 
 passage through the mind, while anything which is 
 fallacious and logically wrong, at once shocks his 
 habits of thought, and is rejected. He is accustomed 
 to a chain of deduction, where each link hangs on the 
 preceding; and thus he learns continuity of attention 
 and coherency of thought. His notice is steadily 
 fixed on those circumstances only in the subject on 
 which the demonstrativeness depends, and thus that 
 mixture of various grounds of conviction which is so 
 common to other men’s minds is rigorously excluded 
 from his. He knows that all depends upon his first 
 principles, and flows inevitably from them; and that 
 however far he may have travelled, he can at will 
 go over any portion of his path, and satisfy himself 
 that it is legitimate; and thus he acquires a just per¬ 
 suasion of the importance of principles on the one 
 hand, and, on the other, of the necessary and constant 
 identity of the conclusions legitimately deduced from 
 them.’* 
 
 Dr. Olinthus Gregory, in his farewell address to the 
 pupils of the Royal Military Academy, speaks of the 
 study of Geometry as leading to that valued attain¬ 
 ment—‘ The power of mastering the mind. If it be 
 desirable to obtain and keep the ascendancy anywhere, 
 it is, surely, at home , in the centre of your own intel¬ 
 lect and its principles, your heart and its emotions. 
 
 * Thoughts on Mathematics as a part of Liberal Educa¬ 
 tion, by the Rev. William Who well, Cambridge. 
 
MATHEMATICS. 
 
 43 
 
 Gain the mastery here, govern within, learn to direct 
 your thoughts to any subject you please, and keep 
 them uninterruptedly to their occupation, till at yovr 
 bidding the labour shall be remitted : then all will be 
 well, for all will lead to a prosperous issue.’ 
 
 We all know, that if human legs made peregrina¬ 
 tions without the consent of the owner, how great would 
 be the hindrance to business, yet how often is thought 
 found roaming the world over when it should be con¬ 
 fined at home But the command of the thoughts is 
 as essential to success in thinking as the control of the 
 limbs is to the dispatch of business. Inasmuch as 
 Mathematics imparts this power, it is valuable dis¬ 
 cipline. 
 
 In the same address Dr. Gregory justly remarks, 
 that the power of doing one thing at one time is the 
 hardest and the most important lesson a young man 
 has to learn. He is of opinion that the concentration 
 of attention and good sense included in this habit, is 
 powerfully impressed upon the notice of the mathe¬ 
 matical student. The wisdom of this course, as ap¬ 
 plicable to children as well as adults, has lately been 
 forcibly illustrated by a lady in these words :—‘ A 
 very quick and clever child made an observation to her 
 governess before me the other day, which had 
 much of truth in it. “ How is it, my dear,” inquired 
 the lady, “that you do not understand this simple 
 thing?” “I do not know, indeed,” she answered, 
 with a perplexed look, “ but I sometimes think I have 
 so many things to learn that I have not time to under¬ 
 stand.” ’ 
 
 If we may trust the testimony of another enthu¬ 
 siast, this science exercises a species of influence 
 of a very high order. ‘ The moral influence of the 
 Mathematics did not escape the penetration of Plato, 
 who, chiefly on that account, decides that the study 
 
44 
 
 INTRODUCTION TO TTIE 
 
 should be cultivated in his “Republic.” “ Geometry,” 
 says he, “ is the knowledge of that which is eternal; 
 it disposes the mind to the contemplation of truth, and 
 gives a philosophical habit, which withdraws our 
 thoughts from what is low 7 to what is elevated. Let 
 not the citizens of this renowned Republic neglect the 
 study of Geometry. ” ’ Mr. Cooley concludes with 
 observing — 1 We well know how much it conduces to 
 the easy acquisition of any kind of knowledge, to have 
 previously learned Geometry.’* 
 
 But it is far from my intention to rest the repu¬ 
 tation of geometrical studies upon declamation, how¬ 
 ever eloquent, or upon testimony how’ever respectable. 
 An exposition of the Logic of Euclid is the best demon¬ 
 stration of the value of its discipline, and this I shall 
 now attempt. 
 
 CHAPTER VI. 
 
 THE I.OGIC OF EUCLID. 
 
 ‘ Among the various guides which have been written for 
 the direction of the understanding, no system of logic has 
 ever been devised better calculated for this purpose than the 
 Mathematics.’— James Harris: ‘Introduction to Compendium 
 of Algebra.’ 
 
 Moderating somewhat the absolute tone of Mr. 
 Harris’s opinion, it may be taken as expressive of the 
 truth. The ancients considered Euclid their best book 
 of lo<ric. Its main features as such are, that from a few 
 axioms, self-evident, and definitions, agreed upon, new 
 truths are evolved by the perception of the coherence 
 
 Cooley’s Introduction to the Study of Mathematics. 
 
MATHEMATICS. 
 
 45 
 
 subsisting between them and the things previously 
 admitted. These truths are added in their turn to the 
 ranks of axioms from which others are to 1 e interred. 
 Until this day sciences are considered perfect in } ro- 
 portion as they possess these features.* To Geometry 
 belong three principal things:—Axioms, or first prin¬ 
 ciples—Definitions, or explanations of the meanings of 
 the chief terms employed—Postulates, or things re¬ 
 quired to he granted. 
 
 An enquirer must understand himself, and disputants 
 must understand each other, or efforts after convic¬ 
 tion will he futile. The geometrician (for we shall use 
 him as the exponent of mathematical procedure) 
 commences by distinguishing what he must assume: 
 for such is the limited nature of human intellect, that 
 all reason must primarily rest on assumption. Ac¬ 
 cordingly, Euclid lays down axioms which, being, as 
 their name implies, self-evident, become the com¬ 
 mon point from which disputants can start on the path 
 of investigation. In geometrical treatises, Definitions 
 are usually placed first, hut in logical propriety 
 Axioms demand first to be considered, as the assent to 
 definitions is often made by a secret reference to 
 axioms in the thoughts. 
 
 AXIOMS. 
 
 The first principles, or fundamental axioms com¬ 
 monly used in Geometry are these :—‘ If a thing be 
 divided into parts, the whole is greater than any one 
 of the parts. Two straight lines cannot enclose a 
 space. All lines drawn from the centre to the circum- 
 
 * Mr. James Milne, in a curious work on National Educa¬ 
 tion, has constructed outlines of Political Economy, Meta¬ 
 physics, and other subjects, on these principles. Though 
 not entirely successful, they are instructive speculations. 
 
46 
 
 INTRODUCTION TO THE 
 
 ference of n circle, are equal to each other. Two things 
 cannot be both equal and unequal at the same time. 
 Things equal to the same thing are equal to one an¬ 
 other.’* The information conveyed by sucli elemen¬ 
 tary truisms is necessary : they are the very A, B, C, 
 of the science—the lowest steps of the mathematical 
 ladder, and as essentially useful as the highest. 
 
 The patient reference which the geometer makes to 
 these first principles is highly instructive. He never 
 loses sight of them. At his highest altitude lie turns 
 to refresh his understanding at those clear fountains of 
 sense—an example which every reasoner may imitate, 
 and an advantage which the investigator in every 
 science may profitably secure to himself. 
 
 Dr. South has encouragingly remarked, that the 
 reason of all things lies in a narrow compass.f The 
 geometrician is professionally master of this secret, so 
 true of every rational theme of human speculation. 
 An Iliad of meaning lies concealed in the nut-shell of 
 a fundamental axiom. Of what importance is this 
 truth to the enquiring public of our day ? 
 
 ‘ How many readers are there who would not be 
 glad of attaining to knowledge the shortest way, see¬ 
 ing the orb thereof is swollen to such a magnitude, 
 
 * It strikingly evidences the intimate connection between 
 Logic and Geometry that this axiom and its converse are 
 found at the root of Aristotle’s Syllogisms. Syllogisms 
 were deducible from these two principles—First: things 
 which agree with the same thing, agree with one another. 
 Second : tilings whereof one docs, and one does not agree 
 with the same thing, do not agree with one another.— Bruce. 
 
 t The reason of tilings lies in a narrow compass. Most of 
 the writings in the world are but illustration and rhetoric, of 
 little consequence to a mind in pursuit after the philosophical 
 truth.— Dr. South. 
 
MATHEMATICS 
 
 41 
 
 and life but such a span to grasp it ? How many w ho 
 have not some curiosity to know the Joundations of 
 those tenets upon which they so securely trust their 
 understanding ? or where the fort steps of those opin¬ 
 ions and precedents may be found, w hich have given 
 direction to so many modern performances l * 
 
 No science is fully mastered till the principles on 
 which it is based are clearly seen. When these are 
 lost sight of, all subsequent reasoning is confused. Tin- 
 geometrical student is early and much accustomed to 
 keep these important considerations before him. The 
 most abstruse proposition is an axiom to the mind 
 that perceives its truth, and the most complicated 
 science resolves itself into a very simple affair when 
 its first principles become familiar to the understand¬ 
 ing. These considerations reduce learning to an art, 
 and give tone, power, and comprehensiveness to tin- 
 understanding, which fit it peculiarly for the acquisi¬ 
 tion of knowledge. 
 
 Ordinary experience in learning and teaching must 
 frequently impress the truth of these remarks on care¬ 
 ful thinkers. When a pupil has a voluminous gram¬ 
 mar put into his hands, lie feels, with Dr. Johnson, that 
 a great book is a great evil. J3ut when the first 
 principles of language are clearly seen, the difficulties 
 vanish and the study is mastered. 
 
 This is not only the secret of the acquisition of know¬ 
 ledge, but also the agent for retaining it. Around these 
 first principles, as around a standard, the thoughts natu¬ 
 rally associate. Touch but a remote chord of any ques¬ 
 tion, and it will vibrate to the central principles to which 
 it has once been well attached—every relative impression 
 owns a kindred connection, and the moment one is 
 attacked, it, like a faithful sentinel, arouses a whole 
 
 * Oldys. 
 
48 
 
 INTRODUCTION TO THE 
 
 troop, who, marshalled and disciplined, bear down 
 with their whole strength, and challenge the enemy. 
 
 DEFINITIONS. 
 
 That confusion in the varying meaning of terms, 
 which reigns through most subjects that agitate 
 mankind, is unknown in Geometry. The Definitions 
 at the very threshold for ever fix the question of sense. 
 Every leading term is defined. There are no words 
 used in Geometry whose meanings are so much 
 alike, that the ideas for which they stand may be con¬ 
 founded together. If there is any distinction between 
 the meanings of two terms, it is a total and a complete 
 distinction. Locke, in illustration of this point, 
 remarks —‘ The idea of two , is as distinct from the 
 idea of three, as the magnitude of the whole earth is 
 from that of a mite ; this is not so in other modes, in 
 which it is not so easy, nor perhaps possible for us to 
 distinguish between two approaching ideas, which yet 
 are really different; for who will undertake to find a 
 difference between the white of this paper, and that of 
 the next degree to it ?’ 
 
 The meanings once fixed in Geometry are ever 
 after adhered to : the same word is for ever used in 
 the same sense. For instance, the term right signifies 
 straight , and straight is its everlasting meaning in 
 Geometry. In political philosophy, right is a well- 
 known stumbling block, which perpetually overthrew 
 the fine genius of Burke. ‘ Natural right ,’ 1 inherent 
 right,’ * moral right,’ and ‘ political right,’ are only a 
 few of the rights which every now and then some 
 puzzled politician will pronounce wrong. It perhaps 
 is not possible to restrict popular phrases as we can 
 scientific terms. But the geometrician, perceiving 
 the advantages of doing so,/acquires the taste of 
 precision, and takes care (vvjnich is practicable with 
 
MATHEMATICS. 49 
 
 everybody) to define the principal terms he may em¬ 
 ploy in any given dissertation. 
 
 POSTULATES. 
 
 With that precision from which the geometer never 
 departs, he defines his mode of* procedure. For this 
 purpose he puts up his Postulates or petitions to be 
 permitted a certain latitude of action. This latitude 
 is never exceeded. Thus neither his language nor 
 his practice is left open to cavil. Admit my axioms, 
 agree to my definitions, and grant my Postulates, 
 says the independent geometer, and I will engage to 
 demonstrate every proposition I lay down. 
 
 A proposition to be proved, or a problem to be 
 solved, have common parts—the Proposition, Enun¬ 
 ciation, Construction, Demonstration, Conclusion. 
 (These parts, for the assistance of the uninitiated, are 
 clearly distinguished in the following pages.) * 
 
 The proposition being plainly laid down, the enun¬ 
 ciation fixes its meaning, the construction provides for 
 its proof, and a rigid demonstration leads the way to 
 the conclusion. Every step has its distinct reference, 
 establishing its legality and fairness. There is no 
 attempt at evasion—no manoeuvre, no equivocation. 
 This daring self-reliance cannot fail to impart manly 
 qualities. 
 
 The student is distracted with no impatient attempts 
 to accomplish two things at once. One thing at one 
 time is the order of Geometry. In the language of the 
 propositions, all circumlocution is avoided. What is 
 intended is straightway expressed. No digressions 
 
 * One who does not understand the principles of Euclid’s 
 Demonstrations, knows absolutely nothing of Geometry: unless 
 he attain this point, all his labour is utterly lost.— Dr. Whatt.y. 
 Preface to Logic,’ p. xxi. 
 
50 
 
 INTRODUCTION TO THE 
 
 
 are endured. Nothing is admitted but that which has I 
 been sanctioned, or established, or is relevant. Nothing ] 
 is asserted but that which is intended to be demon- 1 
 strated. It is impossible that these methods of p ro ! 1 
 cedure can be perceived and practised by the student 1 
 without his acquiring valuable lessons, applicable to 
 other subjects, and to much of the conduct of life. 1 
 The reader, in this place, must be cautioned against 
 the language in which the Logic of Euclid is some¬ 
 times eulogised. Here is an extract from a recent ' 
 journalNever does the mathematician say, such a. 
 proposition is true because it is probable; or, because 
 the balance of the pro and con arguments preponderates 
 in its favour ; or, because history records it, and tradi¬ 
 tion affirms it; or, because the holy St. This declared* 
 it, arid the learned Philosopher That believed it; or, 
 because infallible I assert it. No ! he takes a nobler 
 stand, he claims more lofty ground ; his words are—“ I 
 have actually and incontrovertibly proved so and so to 
 be a fact; disbelieve me, if you can; refute me, if you 
 are able.” ’ This is the folly that encourages the con¬ 
 tempt with which this fine study is sometimes treated.* 
 The reason why the geometer neither refers to 
 probability nor authority is, that the nature of his 
 science enables him to do without them. In the 
 affairs of mankind there are numerous vital questions, 
 such as those of law and legislation, which, from 
 the nature of things, must be decided by probability: 
 by pro and con arguments, and such decisions are as 
 respectable and as salutary, if not quite as safe, as 
 the conclusions of Mathematics. Those speculations 
 
 y ... t It was this thoughtless arrogance which provoked the 
 Iffcll mm nnv«nfr-ovnlmmtioa from a recent critic—‘This logical and 
 
 mathematical school of literature can only hold its sway with 
 narrow minds andin an unimpassioned age .’—Douglas Jerrold't 
 
 i passioned age 
 
 * Shilling Magazine,’ vol. 1, p. 572. 
 
MATHEMATICS. 
 
 *>1 
 
 —and others, curious and important ones, in which 
 authority and personal conviction for a long time 
 form the only accessible evidence—are not to be re¬ 
 jected on that account, but to such evidence is to be 
 allowed its proper weight. Let not flippant arrogance, 
 rejoicing in the strength of Geometry, depreciate the 
 arguments peculiar and necessary to other subjects. The 
 modest mathematician learns the lessons of valuing 
 strict demonstration wherever it can be had, and rightly 
 refuses conjecture where certainty is to be obtained. 
 He knows, if he has general information, that every 
 proposition cannot be proved afler the manner of 
 Geometry, but he has a right to expect that what 
 is attempted to be demonstrated in the same way shall 
 be demonstrated with equal strictness. 
 
 Geometry is theoretical and practical : demon¬ 
 stration, direct and indirect: and reasoning is syn¬ 
 thetical and analytical. Grammar is an illustration 
 of the synthetical ; Chemistry of the analytical. In 
 Grammar we begin with letters—these arc combined 
 into syllables—syllables into words — words into 
 sentences—and sentences into discourses. This is 
 synthesis, or putting together. In Geometry, the 
 ^ gwwd element of the whole superstructure is a point, 
 a point becomes a line, a line a surface: and of these 
 solids are composed. Geometry is synthetical. In 
 Chemistry, water is decomposed and resolved into 
 its elementary principles by means of analysis, or 
 taking to pieces. This is a similar demonstration, 
 though obtained by a method the reverse of the former. 
 There is a third method—namely, Induction, which 
 Hume, with more precision than Locke, calls the 
 doctrine of proofs. Induction is inferring from several 
 particular facts, or propositions, a general truth. All 
 these modes have important uses in different investi¬ 
 gations, Geometrical reasoning is inductive. The 
 
INTRODUCTION TO THE 
 
 52 
 
 Axioms and properties of Definitions are the facts from 
 which first conclusions are deduced. Every pro¬ 
 position established, is an addition to these facts, which 
 become the increased bases of inferences. From 
 coherency between ideas, the geometrical logician pro¬ 
 ceeds to coherency between facts, and from the entire 
 body of facts before him he discovers the coherency of 
 his general inference. Reason is the ability to 
 perceive coherences. Reasoning on the abstrusest 
 theorems is nothing more than after having arrived, 
 step by step, at a remote truth, discovering its connec¬ 
 tion with preceding facts in the same chain. Geo¬ 
 metrical problems alarm the uninitiated, who are not 
 aware that a little attention conquers the simple ones, 
 and a little patience the most difficult. One in the 
 secret thinks them all equally easy, because lie sees 
 they are all equally connected. Bo far as geometrical 
 study affords practice in mastering these consecutive 
 coherences, it may be considered as aiding the de¬ 
 velopment of reasoning power. 
 
 The ‘ First Book of Euclid ’ is a perfect illustration 
 of geometrical logic. It embraces the foundation of 
 •all the subsequent books. They deal with new pro¬ 
 positions and advanced reasoning, but the principle of 
 procedure is the same as in the ‘ First Book.’ So 
 when the inquirer understands that, he comprehends 
 the principle of them all. This ‘ First Book ’ furnishes 
 general principles of reasoning, which will carry the 
 student, without other assistance, through every subject 
 to which his attention can be called. He has only to 
 use them as initiative guides—giving him a taste for 
 strictness in every application—not as fettering him to 
 one species of evidence and inference. The discretion 
 here alluded to is that so well enforced by the author 
 of Hermes. ‘ When Mathematics, instead of being 
 applied to exemplify Logic, comes to supply its place, 
 
MATHEMATICS. 
 
 53 
 
 no wonder if it pass into contempt. For when men, 
 knowing nothing of that reasoning which is universal, 
 come to attach themselves for years to a single species, 
 a species involved in lines and numbers only, they 
 grow insensibly to believe these last as inseparable 
 from all reasoning, as the poor Indians thought every 
 horseman to be inseparable from his horse.’ 
 
 CHAPTER VII. 
 
 SATDBAl GEOMETRY. 
 
 Mr. Macaulay, in his historical Essay on Lord 
 Bacon, has some ingenious remarks on the natural 
 origin of the species of Induction with which Bacon’s 
 name is associated. Similar opinions have been ex¬ 
 pressed respecting the Syllogism of Aristotle. One 
 of the ablest opponents of John Locke has some 
 profitable speculations of the same kind concerning 
 reasoning in general, all bringing Science, Art, and 
 Logic home to nature, and resting their foundation 
 there. This is the most encouraging aspect of learn¬ 
 ing that can be presented to the notice of a pupil. 
 No treatise of an educational nature should be 
 without such a feature. If an inquirer once per¬ 
 ceives that the germs of a desired art are familiar 
 to him, he makes an important step in advance— 
 he gains confidence. He finds that he has only to 
 systematise and extend his actual knowledge. The 
 barrier of exclusiveness is broken down, and he enters 
 the common ground of the initiated. A geometrician 
 before referred to, has furnished some interesting illus¬ 
 trations of this truth with respect to his favourite 
 science, accompanied by not less useful remarks. 
 
54 
 
 INTRODUCTION TO THE 
 
 • Suppose Codrus died, and left his field, nearly a square, 
 to be divided among his four sons, Caius, Gaius, Titius, and 
 Tatius. A little consideration would show that it might 
 be done by drawing lines from the middle of one side to "the 
 middle of the other, as A B, C D, (see fig. 1). Thus there 
 would be four squares of one size for the brothers. The 
 middle point would perhaps be found by stepping along one 
 side, and half the number of steps back. Or with a line or 
 a rod. Such modes would be practical, and such as would 
 be suggested to the simplest mind—for the rudiments of 
 Geometry must be in the human mind before the art itself 
 could be formally known as a system. 
 
 ‘ Suppose the field to be three cornered (see fig. 2). A 
 little more (though not much) sagacity would perceive that 
 its division would be effected by drawing a line from any 
 corner D of the field to the middle point A of the opposite 
 side—and from A to C and B, the middle points of the 
 remaining sides. 
 
 ‘ But if the field had been of the unmanageable form as 
 fig. 3, something more than common judgment would be 
 necessary to subdivide this into four equal portions. Some¬ 
 thing would be wanted beyond what the senses or unculti¬ 
 vated reason could furnish, to recognise the equality of the 
 parts A, B, C, D, when the figure had been accidentally 
 divided as required. Then men would begin to consider 
 how they might determine the equality or non-equality of 
 differently shaped figures. They would,' as we are told was 
 the practice of the early philosophers, draw figures in sand 
 or dust to investigate their proportions: first, because it 
 was useful, aud afterwards, because curious. This would 
 be land measurement in miniature. The next removal 
 would be from the sand to the slate, from the dust to the 
 board, aud from these to parchment or paper. What before 
 engaged peasants without doors, would now employ philo¬ 
 sophers within: what was rude, vague, and unsatisfactory, 
 would now become accurate, definite, and conclusive: what 
 was done experimentally with the rod and the tape, would 
 now be determined rationally with the rule and compass. 
 Behold the gradual transformation of the practical into the 
 abstract art—of primitive into refined Geometry ! 
 
 ‘ Ask. a ploughman how he manages to keep his furrows 
 
MATHEMATICS. 
 
 fK> 
 
 parallel, and he will tell you, by driving his team and 
 directing the share, so that each new furrow shall keep at 
 the. same distance from the last, throughout the whole 
 length. In like manner the school boy rules his copy 
 straight by keeping each end of the ruler at the same dis¬ 
 tance from the last line ruled. Thus is the doctrine of 
 parallels found inherent in the commonest minds. 
 
 ‘ Let us by all means get rid of the scholastic notion, that 
 Geometry is a thing totally distinct from, and independent 
 of, the material world—irrelevant to the ordinary business of 
 life, and uncongenial with the common trains of human 
 reflection : that it is, in fact, an ethereal system, not refined 
 from the grossness of terrestrial conceptions, but absolutely 
 free from all intermixture or affinity with what we sec or 
 know about earthly matters: and that it is drawn from the 
 skies, as Socrates drew his moral philosophy, and is a study 
 for those alone who agree to call each other “ purely in¬ 
 tellectual beings.” No such thing. It is only the simplest 
 and most certain of our ideas, gathered from practice and 
 experience, rendered accurate by strict circumspection, and 
 combined with elegant ingenuity into a system differing 
 from others of human invention only in being more perfect. 
 So far is it from being anything of a supernatural or occult 
 science, that a man of strong judgment might, in a popular 
 way, form to himself a system of Geometry almost without 
 knowing how to read. In fact, a great many illiterate men 
 do so. What a powerful system of natural Geometry must 
 the celebrated Watt have organised in his mind! lie was 
 ignorant of this science (that is, of the written word of it), 
 but he was as well acquainted with the common truths of 
 it as the most deeply initiated mathematician, or he would 
 never have made his great discoveries, which involve those 
 properties.’ ♦ 
 
 A little reflection on the Elements of Geometry, ns 
 displayed in the following pages, will further show 
 that the principles of the science have their rise in 
 human experience. The 1 cutting off the corner’ of a 
 street in walking evinces an intuitive knowledge of a 
 
 * Abridged from Barley’s Geometrical Companion. 
 
56 
 
 INTRODUCTION TO THE 
 
 geometrical theorem. The elements are common to all 
 comprehensions—Science develops and perfects them. 
 
 CHAPTER VIII. 
 
 PRACTICE AND THEORY—OR, THE DISTINCTION BETWEEN 
 PRACTICAL GEOMETRY AND 1*URE MATHEMATICS. 
 
 Pure Geometry is theoretical. Euclid’s Elements 
 are considered pure Mathematics. In this species of 
 Geometry everything is done in accordance with the 
 Postulates of Euclid. No instruments, nor compass, 
 nor rule, are used, and the results arrived at are all 
 abstract and unapplied. 
 
 Practical Geometry, on the other hand, is seen in 
 the constructions connected with the Propositions, and 
 in the * Uses’ annexed to our ‘ First Book.’ In Prac¬ 
 tical Geometry, compasses, sections, rules—all species 
 of instruments are employed. It is the application of 
 Geometry to the arts of life. 
 
 In pure Geometry something of the practical cer¬ 
 tainly appears in the ‘ Constructions ’ of certain of 
 the Problems and Propositions, but is employed no 
 farther than is necessary for the development of 
 the demonstration—the practical is subordinate, while 
 in Practical Geometry it is the essence. Pure Geo¬ 
 metry chiefly demonstrates that a given Proposition is 
 true, or that a given thing can be done. Practical 
 Geometry, on the other hand, teaches all convenient 
 methods of practice, and applies it to any or every¬ 
 thing to which it can be useful. Such is the aptitude 
 of human intelligence for reality, that when we assume 
 to apply only the Postulates of Euclid, we think of 
 the instruments; but this difference still remains—in 
 Practical Geometry the rule and compass are in the 
 hands , in pure Geometry only in the thoughts. 
 
MATHEMATICS. 
 
 In this work it is attempted to exhibit the ‘ Beauties’ 
 of Euclid’s Elements theoretically, and the ‘ Uses’ 
 practically—because where the theory only is ex¬ 
 hibited the question is heard—‘It is all very beau¬ 
 tiful, but what is the use of it ? * If only the 
 practice is shown, the remark arises—‘ I sec it is so, 
 but I do not see why.’* The question and the remark 
 are both proper, and both should be met. These dis¬ 
 tinctions rule throughout Mathematics. Whatever 
 chiefly pertains to demonstration, and shows that a 
 certain thing can be done, or that a certain result will 
 ensue, belongs to the province of pure Mathematics 
 —while in Practical Mathematics attention is mainly 
 devoted to modes of operation and to the application 
 of principles. In the pure department, all is abstraction 
 and theory—in the other, all application and practice. 
 
 CHAPTER IX. 
 
 EXORDIAL ADDRESS TO THE STUDENT. 
 
 More than 2,000 years have rolled away since the entire 
 Elements were given to the world. A halo of interest, 
 peculiar to no other work known among men, surrounds it. 
 It arose in the morning of Science, flourished among the 
 glorious Geometers of old, kept alive the sacred fire of 
 Learning, when almost the whole earth was without one 
 genial ray, and it now shines with undirninished lustre amid 
 the splendour of modern discovery. It has resisted the 
 attacks of countless critics, great and clever men—has sup¬ 
 ported the weight of innumerable commentators ; still it 
 remains the most perfect work of man, and the finest monu¬ 
 ment of his reasoning powers. It has been translated into 
 all languages, has been taught for centuries in every mathe- 
 
 Ritchio’s Principles of Geometry. 
 
INTRODUCTION TO THE 
 
 58 
 
 matieal school of eminence, and is now reputed the best 
 introduction to that science extant. 
 
 It may safely be asserted, that as Grammar is the key ot 
 literature, so is Geometry the key of the sciences. Without 
 exaggeration, it lias been contended that ‘ the only road to 
 accurate and independent knowledge of most of the sciences 
 lies through the study of Mathematics.’* Euclid is the 
 great schoolmaster of the sciences. IJis Elements are the 
 pass-word into their dominions. ‘ Without a sufficient 
 knowledge of Mathematics, that great instrument of [all] 
 exact inquiry, no man can ever make such advances in the 
 higher departments of science as can entitle him to form 
 an independent opinion on any subject within their range.’t 
 
 Amasis w'ondcrcd at the skill of Thales when he mea¬ 
 sured the Pyramids by means of their shadow. Neither the 
 power nor station of Amasis could place him on a level 
 with the distinguished geometer. Ptolemy, another Egyptian 
 king, is said to have asked Euclid if there was no easier 
 mode of becoming a geometrician than by studying his 
 Elements. ‘ There is no royal road,' was his memorable 
 answer. The meaning of which is, that the acquisition of 
 no single idea can be made in a royal way. Only one path 
 was open to Ptolemy and to mankind. Science owns no 
 idle votaries. The only condition on which it yields its 
 secrets is that of attention. 
 
 Do not fall into the error of vanity, and neglect learning 
 under the impression that it courts you. It does indeed in¬ 
 vite you, but it is in the language of independence. It is a 
 benefactor, and never descends to supplicate the accept¬ 
 ance of its favours. It bestows honour and power on those 
 who seek its wealth, but it never yet conferred a gift on any 
 but the earnest. 
 
 Revolve the last words of the venerable professor of 
 Mathematics (Olinthus Gregory)—‘In your mental and 
 scientific pursuits; define to yourselves clearly the purposes 
 which you have in view ; see to it that they are in no 
 way incompatible with the nature and the duty of man, and 
 you cannot direct your aim at too high a point.’ 
 
 Review for the Many. 
 
 ] Sir John Ilerschel. 
 
EUCLID. 
 
 50 
 
 the FIRST BOOK OF EUCLID. 
 
 SECTION I. 
 
 DEFINITIONS. 
 
 1. A Point is that which has no parts, or which has no 
 
 magnitude.* 
 
 2. A Line is length without breadth-t 
 
 Corollary. —The extremities of a line are Points : and 
 the intersection of one line with another is also a Point. 
 [Thus (fig. 4) the place A is a point]. 
 
 3. Right or Straight lines are such that cannot coincide 
 in any two points without coinciding altogether.]; [This is 
 substituted for Euclid’s, that the Corollary may appear 
 more evident]. 
 
 Cor.— Hence two straight lines cannot enclose a space. 
 Neither can two straight lines have a common segment— 
 that is, they cannot coincide in part without coinciding 
 altogether. [The 14th prop, will illustrate this]. 
 
 * A Point is that which has no parts. — Dec/tales. 
 
 A Point is that which has position, but not magnitude.— 
 
 Playfair. 
 
 A Point is the extremity of a line having no dimensions of any 
 kind—neither length, nor breadth, nor thickness. — Gcoin. Lib. 
 
 Useful Knowledge. 
 
 A Point is a mark in magnitude, w'hich is (supposed to be) 
 
 indivisible.— Zeno. 
 
 A Point is the beginning, as it were, of all magnitude, as unity 
 
 is of number.— Taequet. 
 
 A Point occupies no space.— Legendre. 
 
 f A Line has length only, and wants all breadth, inasmuch as 
 it is understood to be produced by the flowing of a point.— Taequet. 
 
 I } A Right line is that whose extremes hide all the rest.— Plato. 
 A Right line is the shortest of all those that can be drawn 
 
 between two points.— Archimedes. 
 
GO 
 
 EUCLID. 
 
 4. Parallel straight lines arc such as arc in the same 
 plane, and which being produced ever so far both ways do 
 not meet.* (See fig. 5). 
 
 5. A Superficies , or surface, is that which has only 
 length and breadth. 
 
 Cor.—T he extremities of a superficies are lines, and 
 the intersections of one superficies with another are also 
 lines.f 
 
 6. A Plane , or plane surface, is that in which any two 
 points being taken, the straight line between them lies 
 wholly in that superficies .I 
 
 7. A Plane Angle is the inclination of two lines to one 
 another, which meet together, but are not in the same right 
 line. 
 
 [The two lines, which, meeting together, make an 
 angle, are called the legs of that angle. A B, A C, (fig. 7) 
 are the legs of the angle B A C. The point A, at which 
 the legs meet, is called the vertex of the angle. An angle 
 may be designated by a single letter, when its legs are the 
 only lines which meet at its vertex. But when more than 
 two lines meet at the same point, in order to avoid confusion 
 three letters are employed to designate the angle at that 
 point—the letter which marks the vertex of the angle being 
 always placed in the middle. Thus (fig. G) the lines G C 
 and E C meeting together at C, make the angle G C E, or 
 E C G: the lines G C, E C, are the legs of the angle, the 
 
 * Lines are Parallel to each other if the perpendiculars be 
 tween them are all equal to each other_ Archimedes. 
 
 Newton, in Lemma 22, Bk. I. of his ‘ Principia,’ says that 
 parallels are such lines as tend to a point infinitely distant. 
 
 f A Superficies has two dimensions, and is understood to be 
 produced by the flowing of a line.— Tacquet. 
 
 | A Plane surface is a surface which is perpetually even, or 
 flat throughout its whole extent.— Darley. 
 
 (On such planes all the figures in the First Book are supposed 
 to be constructed). 
 
 A Solid is a magnitude having the three dimensions of space : 
 length, breadth, and depth. Any one of the six sides of a Cube 
 is a surface, or superficies—an edge is a line —a corner a point. 
 (See fig. 30). 
 
EUCLID. 
 
 01 
 
 point C is the vertex. In like manner may be named the 
 ar,d e E C A, which is the sum of tire angles G C E, G C A: 
 and so of the other angles round the same point. 
 
 When the legs of an angle are produced (or continued) 
 beyond the vertex, the angles made by them on both sides 
 of the vertex are said to be vertically opposite to each other: 
 thus, since G C is continued ft) D, and E C to F, the angles 
 GCE DCF, are said to be vertically opposite to each 
 other.' And so of the other angles on each side C. 
 
 It must be understood, that by an angle is not meant 
 the surface between the lines which form it. For 
 though the surface be increased by producing the legs, (to 
 D and E, for instance, in fig. 7), the angle will still remain 
 the same in magnitude. By an angle, in fact, is meant the 
 degree of width, or separation between the lines that form 
 it: thus, the opening between B A C is greater than the 
 angle FAC.] 
 
 8. When a straight line, standing on another straight 
 line, makes the adjacent angles equal to each other, each of 
 these angles is called a Right Angle , and each of these lines 
 is said to be at right angles to, or perpendicular to the 
 
 other.* (See fig. 8). 
 
 9. An Obtuse angle is that which is greater than a right 
 angle. (See angle C B A, fig. 49). 
 
 10. An Acute angle is that which is less than a right 
 angle. (See angle A B D, fig. 49). 
 
 1J. A Figure is that which is bounded by one or more 
 lines. (See fig. 9). The space enclosed within a figure is 
 called its area. 
 
 12. A Circle is a plane figure bounded by one line 
 called the circumference, or periphery; of which all lines 
 drawn from a certain point within the figure to the circum¬ 
 ference are equal to one another.! (See fig. 10). 
 
 * A carpenter’s square is a ‘ right angle.’ Pythagoras is said 
 to have been the inventor of the Square. 
 
 A perpendicular line inclines neither to the right hand nor to 
 the left. From this circumspect idea probably arose the pre¬ 
 cept of the poet— 
 
 ‘ Far from extremes a middle course is best.’ 
 
 | Till the time of Plato this was the only curve admitted into 
 
 Geometry. 
 
02 
 
 EUCLID. 
 
 1:3. That point is called the Centre of the circle. 
 
 14. A Diameter of a circle is a straight line drawn- 
 through the centre, and terminating on both sides in the 
 circumference. 
 
 15. A line drawn from the centre to the circumference 
 of a circle is called the Radius , or Semi-diameter. 
 
 [ Radii is the plural of radius]. 
 
 1(5. Rectilineal figures are those contained by straight 
 lines. 
 
 17. Trilateral figures, or Triangles, by three straight 
 lines. (See fig. li). 
 
 18. Quadrilateral figures by four straight lines. (See 
 fig. 12). 
 
 19. Of three-sided figures: an Equilateral triangle is 
 that which has three equal sides. (See triangle A C B, 
 fig. 31). 
 
 20. An Isosceles triangle is that which has two equal 
 sides. (See triangle A B C, fig. 37). 
 
 21. A Right-angled triangle is that which has a right 
 angle.* See triangle A E D, fi". 13(5.) 
 
 [The angle opposite the base is called the vertical angle. 
 A side of a triangle, considered in reference to the angle 
 opposite, is said to subtend it, or bo the subtense of the 
 angle. When the side of any triangle is produced, as B C 
 to D, (fig. 14) the angle A C D, made by the produced part 
 C 14, with the other leg A C, is called the external angle. 
 And the angle A C B, adjacent to the external angle, is 
 called the iiiternul adjacent angle. The other two angles ot 
 the triangle are together called the internal remote angles : 
 and of these, that of which the produced side is a leg, 
 namely, A B C, is the internal opposite, and the other, 
 namely, B A C, is the internal alternate angle.] 
 
 22. Of quadrilateral figures : a parallelogram is that of 
 which the opposite sides arc parallel. (See fig. 15). 
 
 23. The straight line which joins the opposite angles of 
 a quadrilateral figure is called a Diagonal. 
 
 * Any side of a rectilineal figure may be called the Rase. In 
 a right-angled triangle (see fig. 13) the side « c, opposite the 
 right angle is called the Hijpolheneuse, either of the other two sides 
 a b, a c, the Base, and the one not taken the perpendicular. 
 
KUCI.II). 
 
 03 
 
 24. The Altitude of a triangle or a parallelogram is a 
 perpendicular drawn from the opposite angle or side, upon 
 the base, as a b, c d. (See figs. 16 and 17.) 
 
 25. Of parallelograms: a Square is that which has all its 
 sides and angles equal. (Sec fig. 97). 
 
 26. An Oblong., or Rectangle is a four-sided figure, that 
 has equal angles but not equal sides. (Sec fig. 23). [When 
 a four-sided figure has equal angles, they are always right 
 angles. Every ‘square’ is a ‘rectangle,’but every ‘rect¬ 
 angle ’ is not a square.’] 
 
 SECTION 11. 
 
 POSTULATES. 
 
 (Things required : from the Latin postulo, to require). 
 
 1. Let it be granted that a straight line'may be drawn 
 from any one point to any other point. 
 
 2. That a terminated straight line may be produced to 
 any length in a straight line. 
 
 3. And that a circle may be described from any centre 
 
 with any radius or interval. 
 
 SECTION III. 
 
 AXIOMS. 
 
 (Authorities, or things having authority : from a Greek word. 
 An Axiom is a self-evident proposition). 
 
 1. Things which are equal to the same things, are equal 
 
 to one another. 
 
 2. If equals be added to equals, the wholes are equal. 
 
 3. If equals be taken from equals, the remainders are 
 
 equal. 
 
 4. If equals be added to uncquals, the sums are unequal. 
 •5. If equals be taken from uncquals, the remainders arc 
 
 unequal. 
 
G4 
 
 EUCLID. 
 
 f>. Things which are doubles of the same thing, are 
 equal to one another. 
 
 7. Things which are halves of the same thing, are equal 
 to one another. 
 
 8. Magnitudes which coincide with one another, (that 
 is to say, which fit together so exactly, that every point of 
 the one lies on some point of the other,) are equal. 
 
 9. The whole is greater than its part. 
 
 10. All right angles arc equal to one another. (Legen¬ 
 dre demonstrates this). 
 
 11. Two straight lines which intersect each other can¬ 
 not be both parallel to the same straight line. 
 
 [AH these Axioms are not required in the First Book, 
 but not being numerous they are inserted as samples of the 
 assumptions of Geometry], 
 
 SECTION IV. 
 
 DEFINITIONS OF TEEMS. 
 
 A Theorem is a truth which becomes evident by a train of 
 reasoning called a Demonstration. 
 
 A Problem is a question proposed which requires a 
 solution. 
 
 A Lemma is a subordinate or minor truth, which is estab¬ 
 lished in order to be employed in the demonstration of a 
 theorem, or in the solution of a problem. 
 
 A Proposition is a portion of science. It is a common 
 name applied indifferently to theorems, problems, and 
 lemmas. A proposition is a simple statement respecting 
 any subject.— Jus. Milne. 
 
 A Hypothesis is a fact assumed without proof. (Thus, 
 when it is affirmed that in an Isosceles triangle the angles 
 at the base are equal, the hypothesis of the proposition is 
 that the triangle is Isosceles, or that its legs are equal). 
 
 A Construction is the addition made to, or change made 
 in, the original figure, by dividing or drawing liues, in 
 order to adapt it to the argument of the demonstration, or 
 the solution of the problem. The condition under which 
 

 
 
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