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Cornell University Library 
 
 arV19509 
 
 Some proofs in elementary geometry / 
 
 3 1924 031 295 862 
 
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Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031295862 
 
SOME PROOFS 
 
 IN 
 
 ELEMENTARY GEOMETRY 
 
 BY 
 
 Pkofessob GEORGE WILLIAM JONES 
 
 OF CORNELL UNIVERSITY ~ 
 
JONES' FIVE-PLACE LOGARITHMS. 
 
 PAGES 
 I. LOGARITHMS OF NUMBEBS, 2-19 
 
 A table of flve-place logarithms of four-flgure num- 
 bers, with differences for a fifth figure. 
 
 11. TRIGONOMBTBIC FUNCTIONS, ' - 30-64 
 
 A table of natural sines, cosines, tangents and co- 
 tangents of angles 0°-180°, to minutes, and of their 
 five-place logarithms, with differences for seconds. 
 
 ni. MINOB TABLES, 3-19 
 
 NaturalLogarlthms; Ten-Place Logarithms; Weights 
 and Measures ; Mathematical Constants; Meridional 
 Parts; Square-Koots; Cube-Boots; Reciprocals; Sines 
 and Tangents of Small Angles. 
 
 JONES' LOGARITHMIC TABLES. 
 
 FOUR-PLACE, SIX-PLACE, AND TEN-PLACE. 
 
 EXPLANATION OF THE TABLES, - 3-11 
 
 I. FOUR-PLACE LOGARITHMS, - 12-U 
 
 A four-place table of logarithms of the natural num- 
 bers 1, 3, 3,..999, with a table of proportional differ- 
 ences in the margin, and of the logarithms of the 
 squares, cubes, square-roots, cube-roots and recip- 
 rocals of the numbers 1, 3, 3,. .99. 
 
 II. POUR-PLACE TRIGONOMETRIC FUNCTIONS, - 15-19 
 A four-place table of logarithms of the six principal 
 trigonometric functions, with differences for min- 
 utes, and of the length of arcs in radians. 
 
 IIL LOGARITHMS OF NUMBERS, . 20-37 
 
 A six-place table of logarithms of four-figure num- 
 bers, with a table of differences. 
 
 IV. CONSTANTS OF MATHEMATICS AND OF NATURE 
 
 —WEIGHTS AND MEASURES, - - 38-41 
 
 A table of useful constants, with the logarithms of 
 those in common use. 
 
 V. ADDITION-SUBTRACTION LOGARITHMS, - 43-58 
 
 A six-place table of logarithms so related that, by 
 their use, the logarithm of the sum and of the diff- 
 erence of two numbers may be found from their 
 logarithms without taking out the numbers them- 
 selves. 
 
 VL SINES AND TANGENTS OF SMALL ANGLES, - 69 
 
 A table of the ratios sinA":A, tanA":A for angles 
 0°-5'', whereby the logarithmic sines and tangents of 
 these small angles are found mord exactly than by 
 Table VII. 
 
 (SEE THE THIRD COVER PAGE.) 
 
SOME PROOFS IN ELEMENTARY GEOMETRY. 
 
 The proofs here given are such proofs as I have been 
 accustomed to give in my own classes. They are written 
 out for my pupils; but if any teacher finds them useful I 
 shall be very glad. They are alternative proofs that may 
 be used instead of proofs by limits. They are meant to be 
 rigorous. G. yv. j. 
 
 In this paper the sign •.• stands for iecause, -•. for there- 
 fore, i= for not equal to, = for approaches, meaning there- 
 by that if one magnitude approaches another magnitude 
 of the same kind, they come at last to differ by less than 
 any assigned magnitude, and tend toward equality. 
 
 By a lemma is meant a theorem that is assumed to have 
 been proved, and which is needed in the proof of the 
 theorem under consideration. 
 
 By a ratio is meant that number by which the consequent 
 of the ratio is to be multiplied to give the antecedent. 
 
 By the product of two lines is meant the rectangle of 
 wiiich the two lines are adjacent sides. If they be broken 
 or curved, they are to be straightened first. 
 
 By the product of a line and a surface is meant the 
 rectangular parallelepiped whose altitude is the given line, 
 and whose base is a rectangle equal in area to the given 
 surface. If this surface be broken or curved, it is to be 
 flattened first. 
 
 By a line is meant a straight line. If a line be broken 
 or curved, it is so named. 
 
 By tivo equal lines is meant two lines that are equal in 
 length. 
 
 By two equal surfaces is meant two surfaces that are 
 equal in area. 
 
 By tiuo ecpial solids is meant two solids that are equal 
 in volume. 
 
 1 
 
Theob. 1. If three parallel lines cut ttvo other lines, the 
 like segments of the tivo lines. are proportional. 
 
 Lemma. If three or more parallel lines cut two other 
 lines, and if one of these lines be cut into equal parts, so 
 is the other. 
 
 Let one of the lines be cut by the parallels in a, b, c, and 
 the other in A^, b', g' ; then will ab/bc = k'b'/b'c'. 
 
 (1) AB, BC commensurahle. 
 
 B' 
 
 lA 
 
 -_ 
 
 ------\-- 
 
 c 
 
 c\ 
 
 Let the common measure of ab, bc go into ab m times 
 and into bc n times, and through the points of 
 division draw lines parallel to the given parallels ; 
 • AB is cut into m equal parts, so is aV, [lem. 
 
 ■ BC is cut into n equal parts, so is b^'c', 
 . AB is got from bc by dividing bc into n equal parts 
 
 and taking m of these parts, 
 A^B^ is got from B^c^ in the same way ; 
 AB has the same relation to bc as a^b^ has to b^g^, 
 ab/bc = k.''s,' /-^'g'. q. e. d. 
 
 if wi = 5 and ?2 = 3 ; 
 
 ■ AB is got from bc by multiplying bc by |, 
 A^B^ is got from b^c^ by multiplying B^c' by |, 
 
 .-. ab/bc = K'W/s'c'. Q. E. D. 
 
 (2) AB, BC not commensurahle. 
 Let BC be cut into n equal parts, and let AB contain more 
 than m of these parts and less than wi + 1 of them. 
 Draw lines through these points of division parallel to 
 
 the given parallels ; 
 then •.• B^C is thus cut into n equal parts and A^b' contains 
 more than m of these parts and less than in + 1 
 of them, [lem. 
 
 then 
 and 
 
 and 
 
 i.e. 
 
 and 
 
 e.g. 
 
 then 
 
 and 
 
cV "TcT c^ 61 c^ 
 
 . • . m/n < ab/bc < (m + 1 )/n 
 and ?w/«<aV/bV<(7?2 + 1)/?j; 
 
 .•. there are four ratios Whereof ni/n is least, 
 
 ()K + l)/?i is greatest, and the constant ratios 
 ab/bc and hJ's,' /^'c' lie between them. 
 If possible let ab/bc i^ k!-&' /s.'g' ; 
 then •.• whatever the difference of the constant ratios 
 
 ab/bc and a^b'/^'c', that difference is constant, 
 and •.• the ratios m/n and (;m+l)/n are variables such that 
 their difference l/?i approaches when n is made 
 very great, 
 .•. the least of the four ratios can be made to differ 
 from the greatest of them by less than the two 
 intermediate ratios differ ; which is absurd ; 
 .-. the ratios ab/bc and a^bYbV do not differ, and 
 they are equal. Q. B. d. 
 
 e.g. if BC be the side of a square and ab its diagonal ; 
 then-.- ab/bc =V3 = 1-4142,. .. 
 
 and 1<V2<3, 14/10 <V3<15/10, .. . 
 
 14142/10000<v' 2 < 14143/10000, . . . 
 m/n<\/2< {m + l)/n, wherein w is some very 
 great number, and m is still greater ; 
 .-. 1/??., the difference of the variable ratios m/n and 
 {m + l)/n, can be made smaller than any given 
 fraction when n is made great enough. 
 If, then, the difference of the constant ratios ab/bc and 
 A^ByB^c' be a millionth or any other fixed fraction, 
 the difference of the ratios m/n and (m + l)/n can 
 be made smaller than this fraction, say a billionth; 
 and the absurdity is apparent. 
 3 
 
Theok. 2. Two rectangles that have equal bases are pro- 
 portional to their altitudes. 
 
 Lemma. Two rectangles that have equal bases and the 
 same altitude are equal. 
 
 Theor. 3. Two angles at the center of a circle are pro- 
 portional to their subtending arcs. 
 
 Lemma. If two angles at the center of a circle be 
 equal, their subtending arcs are equal. 
 
 Note. — The proofs of theors. 3, 3 are like that of theor. 1. 
 
 Theoe. 4. If there be two similar regular polygons, the 
 one inscribed in a circle and the other circmnscribed about 
 it, and if the number of the sides be doubled again and again, 
 so that the polygons remain regular and similar, then (1) 
 the difference of their perimeters may be made less than any 
 assigned line andthat of their surf aces less than any assigned 
 surface ; (3) the surface of a circle is half the product of the 
 perimeter of the circle and its radius. 
 
 Lem 1. If two polygons be similar, their perimeters are 
 proportional to any two like lines, and their surfaces are 
 proportional to the squares of such lines. 
 
 Lem. 2. The surface of a regular polygon is half the 
 product of its perimeter and apothem. 
 
 Lem. 3. If there be two polygons, the one inscribed 
 in a circle and the other circumscribed about it, the per- 
 imeter of the circle is greater than that of the inner poly- 
 gon and less than that of the outer polygon ; and so of 
 their surfaces. 
 
 Lem. 4. The difference of two sides of a triangle is less 
 than the third side. 
 
 Let OA be a circle, r its radius, c its perimeter, c its sur- 
 face. 
 Let AB be the side of an inscribed square, and A^B^that of 
 a circumscribed square; and let the number of sides 
 of both polygons be doubled again and again, so 
 that the polygons remain regular and similar, and 
 the sides grow shorter and shorter. 
 4 
 
Let a be the apothem of the inner polygon, p its per- 
 imeter, and s its surface; and let p be the perimeter 
 of the outer polygon and s its surface. 
 
 then 
 
 and 
 
 and 
 
 So 
 
 and 
 
 So 
 
 and 
 
 v/p — r/a, [lem. 1. 
 
 (p —p)/{r — a) = p/r J [proji'n. 
 
 r — rt<halfof one of the short sides of the inner 
 polygon, and this side approaches 0, [lem. i. 
 
 ?■ — «=0 ; 
 
 p<8r and T—p<S(r — a), 
 .-. p— jij = 0. 
 •.• s/s = rya\ 
 .-. (s-s)/(r'-a') = s/r' ; 
 •.•s<4r' and ?•" — «" = (»• + «)(?• — rt) = 0, 
 
 ■.• a<r and p<c<'P, 
 
 Q. B. D. 
 
 [lem. 1. 
 
 Q. E. D. 
 
 [lem. 3. 
 [ax. ineq. 
 [lem. 3, 2. 
 
 .•.s<c<s and s = ^pa, s, = \vr, 
 
 . ■ . there are four surfaces whereof s and its equal 
 
 ipa is least ; s and its equal j-j^r is greatest ; and 
 
 the constant surfaces o and -Jcr lie between them. 
 
 If possible let c ^ icr ; 
 
 then -.• whatever the difEerence of these two constants, 
 
 that difEerence is constant, and the two variables 
 
 s, s can be made to differ by less, [above. 
 
 .-. the least of the four surfaces can. be made to differ 
 
 from the greatest of them by less than the two 
 
 intermediate surfaces differ; which is absurd; 
 
 . • . these two surfaces do not differ, and they are equal. 
 5 
 
Theor. 5. Two rectangular parallelopipeds tliat have 
 equal bases are proportional to their altitudes. 
 
 Lemma. Two rectangular parallelopipeds.that have equal 
 bases and the same altitude are equal. 
 
 Theor. 6. Ttvo diedral angles are proportional to their 
 2}lane angles. 
 
 Lemma. If two diedral angles be equals their plane 
 angles are equal. 
 
 Theor. 7. Tiuo lunes on the same sphere are proportional 
 to their angles ; and so are two such spherical ivedges. 
 
 Lemma. If two lunes on the same sphere be equal, 
 their angles are equal ; and so if two such spherical wedges 
 be equal. 
 
 Note. — The proofs of theors. 5, 6, 7 are like that of theor. 1. 
 
 Theor. 8. If tioo triangular pyramids have the same 
 altitude and bases that are equal in area, the pyramids are 
 equal in volume. 
 
 Lem. 1. If two pyramids have the same altitude and 
 bases that are equal in area, sections made by planes par- 
 allel to the bases and at equal distances from the vertices 
 are equal in area. 
 
 Lem. 2. If two triangular prisms have the same alti- 
 tude and bases that are equal in area, the prisms are equal 
 in volume. 
 
 A„„ „ , „A' 
 
 Y Y 
 
 Let the two triangular pyramids p, p-' have the same alti- 
 tude and their bases xyz, isJ^'i' be equal in area ; 
 
 then are p, p^ equal in volume. 
 
 6 
 
For, cut the edges ax, a^x^ into the same number of equal 
 parts at B, c, . . . b',& . . . and through these points 
 of division draw planes parallel to the bases; 
 then •-• the sections at b, b^ are equidistant from the ver- 
 tices A, A^, 
 .•. they are equal in area; [lem. 1. 
 
 and so are the sections at c, c^ . . . 
 
 On the base xyz, and on the sections at b, c, . . . construct 
 prisms having one edge on ax and reaching each to 
 the section above it. 
 Call these the outer jjrisvis of p. 
 Below the sections as upper bases construct like prisms, 
 
 and call them the inner prisms of p. 
 So construct the sets of outer, and of inner, prisms of ?' ; 
 then •.•AX, A''x^ are cut into the same number of equal 
 parts, 
 ■. all the prisms have the same altitude ; 
 and •.• the two outer prisms on xyz, x^t^z^ have the same 
 altitude and bases that are equal in area, 
 .•. these two prisms are equal in volume, [lem. 2. 
 
 and so are the others, two and two, to the top of 
 both pyramids, 
 .•. the set of outer prisms of p is equal in volume to 
 the set of outer prisms of v' ; 
 and so are the two sets of inner prisms equal in volume. 
 And •-• the first outer prism of p is equal to the first inner 
 prism, counting from the top, the second to the 
 second, and so on, [lem. 2. 
 
 .•. the sum of the outer prisms of p exceeds the sum 
 of the inner prisms by the outer prism at the 
 bottom ; 
 and so for the outer and inner prisms of v'. 
 Let the edges ax, a^x-' be cut into more and more equal 
 parts, so that the plates (prisms) are of equal 
 thickness but growing thinner and thinner. 
 7 
 
Let Y, T^ stand for the sets of outer plates of p, v^, and 
 
 V, v' for the sets of inner plates ; 
 then •.■ y = v/ and v = v' , [above, 
 
 and T, ?; are variables such that v — ?)=0, and so are 
 T^, v' , [above, 
 
 and •.• i»<p<v and 2;/<p/<Y''^ 
 
 .-. there are four solids whereof v and its equal v' is 
 least ; T and its equal v^ is greatest ; and the con- 
 stants, the pyramids p, v' , lie between them. 
 If possible let p ^ s'; 
 
 then •.• whatever the difference of these two constants, 
 that difference is constant, and the two variables 
 T, V, can be made to difEer by less, 
 .•. the least of the four solids can be made to difEer 
 from the greatest of them by less than the two 
 intermediate solids differ ; which is absurd ; 
 -•. the two pyramids do not diifer, and they are equal. 
 
 Note. — It is to be observed that the inequalities t)<p<v and 
 f'<p'<v' above are independent of each other, and are in no man- 
 ner derived one from the other. This independence holds vpith all 
 the pairs of inequalities that occur in the later proofs, and should be 
 emphasized in giving the proofs. 
 
 Theor. 9. The lateral surface of a cylinder of revolu- 
 tio7i is the 'product of its altitude and the perimeter of its 
 base ; and its volume is the product of its altitude and the 
 surface of its lase. 
 
 Lemma. The lateral surface of a right prism is the 
 product of its altitude and the perimeter of its base ; and 
 its volume is the product of its altitude and the sui^face 
 of its base. 
 
 Let OP-A be a cylinder of revolution, op-ab ... a regular 
 inscribed prism, and op-A''b' ... a regular circum- 
 scribed prism having the same number of faces, and 
 let the number of faces be doubled again and again, 
 so that the bases remain regular and similar poly- 
 gons and the faces of the prisms become very narrow. 
 8 
 
Let p be the perimeter of the inner base, p that of the 
 outer base, c that of the circle ; let s be the lateral 
 surface of the inner prism, s that of the outer prism, 
 cyl. that of the cylinder ; and let li be the altitude 
 of the cylinder and of the prisms ; 
 
 then •.• V, p are variables such that p— /^=0, [theor. 4. 
 
 .•. //p, hp are variables such that /<p — /yj=0 ; 
 
 and -.■ s = hp, s = /iP, and p<c<v, [lem. 
 
 and lti[)<lic<liv and s<cyl.<s, [ax. ineq. 
 
 .-.there are four surfaces whereof s and its equal 
 hp is least ; s and its equal 7ip is greatest ; and 
 the constants ct/l. and Jic lie between them. 
 
 If possible let cyl. i= he ; 
 
 then •.• whatever the difference of these two constants, 
 that difference is constant, and the two variables, 
 7tp, Ap can be made to differ by less, [theor. 4. 
 
 .•. the least of the four surfaces can be made to differ 
 from the greatest of them by less than the two 
 intermediate surfaces differ ; which is absurd ; 
 
 .•.the sui'face of the cylinder does not differ from 
 the product of the altitude and the perimeter of 
 the base, and they are equal. q. e. d. 
 
So let i be the area of the iuner base, b that of the outer 
 
 base, c that of the circle ; and let v be the volume of 
 
 the inner prism, v that of the outer prism, cyl. that 
 
 of the cylinder ; 
 
 then •.• B, b are variables such that b — J=0, [theor. 4. 
 
 .-. liB, lib are variables such that hB — hb=0 ; 
 and •-■v — hb v = 7iB, and b<c<B, [lem. 1. 
 
 and Jib<Jic<hB and v<cyl.<,y, 
 
 .•. there are four solids whereof v and its equal hb 
 is least; v and its equal liB is greatest ; and the 
 constants cyl. and he lie between them. 
 If possible let cyl. ^ lie ; 
 
 then •.• whatever the difference of these two constants, 
 that difference is constant, and the two variables 
 lib, hs can be made to differ by less, 
 .-. the least of the four solids can be made to diffei* 
 from the greatest of them by less than the two 
 intermediate solids differ ; which is absurd ; 
 .•.the volume of the cylinder does not differ from 
 the product of the altitude and the area of the 
 base, and they are equal. 
 
 Note. — For the lateral surface of a cylinder whose right section is 
 a circle, the proof is substantially the same as lor the surface of a 
 cylinder of revolution ; and so is that for the volume of an oblique 
 cylinder with circle base. 
 
 Theor. 10. T7ie lateral surface of a cone of revolution 
 is half the product of its slant height and the perimeter of 
 its base ; and the volume of such a cone is a third of the 
 product of its altitude and the area of its base. 
 
 Lem. 1. The lateral surface of a regular pyramid is 
 half the product of its slant height and the perimeter of 
 its base ; and the volume of such a pyramid is a third part 
 of the product of its altitude and the area of its base. 
 
 Leji. 2. If the lines a, b be so related to the lines a', b' 
 that in length a approaches a', and b approaches ¥, then, 
 in area, the rectangle a-b approaches the rectangle a'-V. 
 
 10 
 
Lem. 3. If a perpendicular and two oblique lines be 
 drawn from a point to a plane, that line which meets the 
 plane further from the foot of the perpendicular is the 
 longer. r 
 
 Let OP-A be a cone of revolution, op-ab ... a regular 
 inscribed pyramid, and op-a^b^ ... a regular cir- 
 cumscribed pyramid having the same number of 
 faces, and let the number of the sides of the bases 
 be doubled again and again, so that the bases re- 
 main regular and similar polygons and the faces of 
 the pyramid become very narrow. 
 
 Let p be the perimeter of the inner base, p that of the 
 outer base, c that of the circle ; let s be the lateral 
 surface of the inner pyramid, S that of the outer 
 pyramid, cone that of the cone ; and let I be the 
 slant height of the inner pyramid, L that of the 
 outer pyramid and the cone ; 
 L is constant and p, p, I, are variable lines such 
 that p— /)==0 and l — Z=0, [theor. 4, lem. 3. 
 LP, Ip are variable rectangles such that lp — /^j=0; 
 
 • Z<L and p<c<v, [lem. 3, theor. 4. 
 \lp<\-i.c<\\J?\ [ax. ineq. 
 
 • s<cone<s, 
 there are four surfaces whereof s and its equal 
 
 ilp is least ; s and its equal -Jlp is greatest ;, 
 and the constants cone and |lc lie between them. 
 11 
 
 then ' 
 
 and 
 
 and 
 
If possible let cone ^ ii,c ; 
 
 then •-• whatever the difference of these two constants, 
 that difEerence is constant, and the two vari- 
 ables ^Ip and -Jlp can be made to differ by less, 
 .•. the least of the four surfaces can be made to differ 
 from the greatest of them by less than the two 
 intermediate surfaces differ ; which is absurd ; 
 .•. the surface of the cone does not differ from half 
 the product of its slant height and the perim- 
 eter of its base ; and they are equal. Q. e. d. 
 XoTB. — For the volume of the cone, the proof is like that for the 
 volume of the cylinder. 
 
 Theoe. 11. The surface of a frustum of a cone of rev- 
 olution is half the jJroduct of its slant height and the sum 
 of the perimeters of its bases j and the volume of such a 
 frustum is a third of the product of its altitude and the 
 sum of its bases and a mean jjroportional between them. 
 
 Lem. 1. If two cones of revolution be similar, the pier- 
 
 imeters of their bases are proportional to their slant heights; 
 
 and the areas of their bases to the squares of their altitudes. 
 
 Lem. ^. If three magnitudes of the same kind, a, I, c, 
 
 be so related that the ratios a/b, b/c are equal, then is the 
 
 ratio a/c the square of the ratio a/b and of the ratio b/c. 
 
 Let L be the slant height of the cone, I that of the top 
 
 cut off, 1,-1 that of the frustum ; and let p be the 
 
 perimeter of the base of the cone,^; that of the top ; 
 
 then is \{L — l){'P-\-p) the surface of the frustum. 
 
 For •.• v/p = i./l, [lem. 1. 
 
 . . lv=-Lp, 
 and J(L-0(P+7-')=i(LP-f-L;)-Zp-/7j) = i-(LP-?p); 
 
 "and ■.• -^LP is the surface of the whole cone and \lp that 
 of the top, 
 
 .-. \{LV — lp) is the surface of the frustum, 
 
 and |(l - 1) (p -f-p) = i(LP - Ip), the surface sought. 
 
 Q. E. D. 
 
 13 
 
So let H be the altitiide of the cone, li that of the to^j, 
 H — /t that of the frustum, b the base of the cone, b 
 that of the top, and Ji a mean proportional between 
 B and b; 
 then is ■J-(h — 7i) (b + 5 4-m) the volume of the frustum. 
 For ■.•B/h^s'/li', [lem. 1. 
 
 and B/M = M/i, [del mean propor'l. 
 
 .•.B/jr = H/7i and -si/b^u/h, [lem. 2. 
 
 .-. 7«B = HM and Inv — ub; 
 and •.- ■J(h-70 (b + 5 + m) 
 
 = "Khb + h5 + HJi - 7tB - 7*5 - /ni) = -J(hb - lib), 
 and -Jhb is the volume of the cone and ^hb that of the 
 top, 
 -•. ■J(hb — 7«5) is the volume of the frustum, 
 and -J(h — 7i) (b + 5 + m) = i(HB — hb), the volume sought. 
 
 Note 1. — The statement and proof of this theorem apply without 
 change to the frustum of a pyramid. 
 
 Note 3. — So far as concerns the volume of the frustum, the state- 
 ment and proof of the theorem apply to any cone with circle base. 
 
 Theoe. 12. The surface of a zone is the product of its 
 altitude and the perimeter of a great circle of the sphere ; 
 and the volume of the spherical sector, lohose base is this 
 zone, is a third of the product of the surface of the zone 
 and a radius of the sphere. 
 
 Lem. 1. If an isosceles triangle be revolved about a 
 line lying in its plane and passing through its vertex 
 but not crossing the triangle, the surf ace generated by the 
 base is the product of the projection of the base on the 
 axis and the perimeter of a circle whose radius is the per- 
 pendicular from the vertex to the base ; and the volume 
 of the ring generated by the triangle is a third of the prod- 
 uct of this perpendicular and this surface. 
 
 Lbji. 2. If an arc of a circle, its chord, and a broken line 
 made np of the tangents at its ends, be revolved about a 
 diameter of the circle that does not cut the arc, the surface 
 
 13 
 
generated by the arc is greater than that generated by the 
 chord and less than that generated by the broken line. 
 
 Lem. 3. If the line a be so related to the line a' that in 
 length a approaches a', and the rectangle 5 • c be so related 
 to the rectangle ¥ ■ c' that in area i ■ c approaches V ■ d , then 
 in volume the rectangular parallelepiped a • 5 • c approaches 
 the rectangular parallelepiped a'-V-c'. 
 
 Let be the center of a circle, OA. its radius, ak an arc, 
 PQ a diameter not cutting ak. 
 
 Divide ak into a great number of equal arcs ab, bc . . . ; 
 draw the chords ab, bc . . . ; and draw a^b^, b^C'' . . . 
 parallel to ab, bo . . ., tangent to the arc ak, and cut 
 ofE by the radii oa, ob, oc . . ; 
 
 then the isosceles triangles oab, obc . . . are all equal, and 
 so are oa^b'', q-r'c' . . . 
 
 Draw the perpendiculars ax, by, cz . . . Ku, A^x^, k'u^ to 
 PQ, and revolve the whole figure about pq ; 
 
 then three surfaces of revolution are generated : 
 the zone by the arc ak, 
 
 the inner surface by the broken line of chords, 
 the outer surface by the broken line of tangents ; 
 
 and three solids of revolution : 
 
 the spherical sector bounded by the zone and the 
 two cones generated by the radii OA, OK, 
 14 
 
the inner solid bounded by the inner surface and 
 
 the cones, 
 the outer solid bounded by the outer surface and 
 
 the cones. 
 
 (1) The surface of the zone ak is the product f its alti- 
 tude xiT and the 2Krimeter of a great circle. 
 For •.• surf. AB = XY- jDer.oJi, [lem. 1. 
 
 surf . BC = iz • per.OM, 
 
 .-. surf. A. . . K = xu-per.OM, 
 i.e. the inner surface = prod. xu- per. OM. [adding. 
 
 So surf. A'' . . . K'' = x^U'' ■ per. OA, 
 
 I.e. the outer surface = prod. x^u''- per.OA ; 
 and •-• OM^OA and x/u''=xu when the number of 
 arcs is doubled again and again, 
 -•. the products xu-per.oii and x^u''-per.OA are vari- 
 ables such that their difference approaches 0. 
 
 [theor. 10, lem. 2. 
 And •.• the inner surface < zone AK<the outer surface, 
 
 [lem. 2. 
 and prod, xu • per. on < prod, xu ■ per.o a 
 
 <prod. x^u^ • per. oa, 
 
 .•. there are four surfaces whereof the inner sur- 
 face and its equal xu-per. ojr is least; the 
 outer surface and its equal x^u^-per.OA is great- 
 est ; and the constant surfaces, zone ae and 
 prod.xu- per.OA, lie between them. 
 If possible let zone ak i= prod.xu -per.OA ; 
 then •.• whatever the difference of these two constant sur- 
 faces, that difference is constant, and the two 
 variables can be made to differ by less, 
 
 .-. the least of the four surfaces can be made to differ 
 from the greatest of them by less than the two 
 intermediate surfaces differ ; which is absurd ; 
 
 -•, zone AK does not differ from prod.xu- per. OA, and 
 they are equal. Q. e. d. 
 
 15 
 
(3) The volume of the spherical sector o-ak is a third 
 of the product of the zone ak and the radius oa. 
 
 For ••• solid o-AB = ^OM' surf. ABj [lem. 1. 
 
 solid o-Bc = -Jo Ji • surf, bc^ 
 
 .•. solid o-A . . . K = ^OM-surf.A . . . K, [adding. 
 
 i.e. the inner solid = prod. Jom and the inner surface. 
 
 So solid O-A^. . . K'' = -|OA-surf.A/ . . . K^, 
 
 i.e. the outer solid = prod. 4oA and the outer surface; 
 
 and ■.• oii = oA, and the inner surf ace = the outer surf ace, 
 
 .•. the products ^OM by the inner surface and ^OA by 
 the outer surface are variables such that their 
 difference approaches 0. [lem. 3. 
 
 And •.• the inner solid<sph. sect. o-AK<the outer solid, 
 
 [lem. 2. 
 and prod.^OM by the inner surface < prod. ^0 A • zone ak 
 <prod.^OA by the outer surf ace, 
 
 .•.there are four solids whereof the inner solid and 
 its equal prod.^oii by the inner surface is least ; 
 the outer solid and its equal prod. -JOA by the 
 outer surface is greatest ; and the two constant 
 solids sph.sect.o-AK and prod.JoA by zone ak 
 lie between them. 
 
 If possible let sph.sect.o-AK ^ prod.^OA by zone ak ; 
 
 then •.• whatever the difference of these two constants, 
 that difference is constant, and the two variables 
 can be made to differ by less, 
 
 .-. the least of the four solids can be made to differ 
 from the greatest of them by less than the two 
 intermediate solids differ ; which is absurd ; 
 
 .". sph.sect.o-AK does not differ from the product 
 ^OA by zone ak, and they are equal. Q. e. d. 
 
 16 
 
PAGES 
 
 VII. TEIGONOMBTEIC FUNCTIONS, 60-104 
 
 A five-place table of natural sines, cosines, tangents, 
 and cotangents of angles C-ISO", to minutes, and a 
 six-place table of their logarithms, with differences 
 of logarithms for seconds expressed in units of the 
 sixth decimal place. 
 
 VIII. NATURAL LOGARITHMS, . . - - 105-117 
 
 A six-place table of natural logarithms of the deci- 
 mal numbers, .01, .03, .03,. .9.99, of the natural num- 
 bers 1, 2, 3,. ,1218, and of the prime numbers between 
 1218 and 10000. 
 
 IX. PRIME AND COMPOSITE NUMBERS, - - 118-137 
 
 A table of prime and composite numbers from 1 to 
 20000, with the factors of the composite numbers 
 that are not divisible by 3 or 5, and ten-place logar- 
 ithms of the primes. 
 
 X. SQUARES, - 138-139 
 
 A table of the squares of the natural numbers 
 1, 2, 3,..999. 
 
 XI. CUBES, ....--- 140-Ul 
 
 A table of the cubes of the decimal numbers .1, .3, 
 .3,..99.9. 
 
 XIL SQUARE-ROOTS, - - - 142-145 
 
 A table of the square-roots, to four decimal places, 
 of the natural numbers 1, 2, 8,. .999, and of the deci- 
 mal numbers .1, .2, .3,..99.9. 
 
 XIII. CUBE-ROOTS, 146-151 
 
 A table of the cube-roots, to four decimal places, of 
 the natural numbers 1, 2, 3,..999~of the decimals .1, .3, 
 .3,..99.9, and of the decimals .01, .03, .03,..9.99. 
 
 XIV. RECIPROCALS, - - 152-153 
 
 A table of the reciprocals of the decimal numbers 
 .01, .03, .03,..9.99. 
 
 XV. QUARTER-SQUARES, - - 154-157 
 
 A table of the quarter-squares of the natural num- 
 bers 1, 2, 3,..30O0. 
 
 XVI. BESSEL'S COEPEICIENTS, - 158 
 
 A table of BessePs coefficients for second, third, 
 fourth, and fifth differences, for interpolation. 
 
 XVII. BINOMIAL COEFFICIENTS, - - 159 
 
 A table of binomial coefBoients for second, third, 
 fourth, and fifth differences, for interpolation. 
 
 XVIIL ERRORS OF OBSERVATION, - - - 160 
 
 A table of ordinates of the probability-curve, values 
 of probability integrals, and other values. 
 
Jones^ Mathematical Text-Books. 
 
 PRICE LIST, JANUARY 1, 1904. 
 
 I. A DRIIiL-BOOK IN TRIUONOMETRY. 
 
 FOURTH EDITION. 
 
 For high-school and college classes. 
 
 I20m, cloth, xvi + 192 pp. Single copies by mail, $1.00. 
 All cash orders (carriage at the buyer's cost), 75 cents. 
 
 II. A DRIIiL,-BOOK IN ALiGEBRA. 
 
 FIFTH EDITION. 
 
 For high-school seniors and college freshmen. 
 
 i2mo, cloth, xvi + 272 pp. Single copies by mail, $1.00. 
 
 All cash orders (carriage at the buyer's cost), 75 cents. 
 
 AN ANSWER BOOK, for teachers only, 25 cents. 
 
 A BOX OF QUESTION CARDS, for the class-room. $1.00. 
 
 III. liOGARITHDIIC TABLES. 
 
 NINTH EDITION. 
 
 Eighteen tables : (four-place, six-place, and ten-place), with explana- 
 tions; for use in the class-room, the laboratory, and the office. 
 
 Royal 8vo, cloth, 160 pp. Single copies by mail, $1.00. 
 
 All cash orders (carriage at the buyer's cost), 75 cents. 
 
 IV. FOUR-PLACE LOGARITHMS. 
 
 FOURTH EDITION. 
 
 Two tables: one of the logarithms of three-figure numbers, the other 
 of trig;onornetric ratios, and their logarithms, for angles differing by 
 ten minutes. 
 
 i2mo, paper, 8 pp. Single copies by mail, 5 cents. 
 
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 V. FIVE-PLACE LOGARITHMS. 
 
 FIRST EDITION. 
 
 Eleven tables: one of the logarithms of four-figure numbers, one of 
 trigonometric ratios, and their logarithms, for angles differing by minutes ; 
 and nine minor tables. 
 
 Royal 8vo. cloth, 64 pp. Single copies by mail, 60 cents. 
 
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 VI. SOME PROOFS IN ELEMENTARY GEOMETRY. 
 
 FIRST EDITION. 
 
 Twelve theorems. Alternative proofs that may be used instead of proofs 
 by limits 
 
 i2mo, paper, 16 pp. Single copies by mail, 10 cents. 
 
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 Single copies of these books are sent free to teachers of mathematics for 
 inspection For the most part they follow well-worn lines; but in some 
 things there are radical departures; and teachers are advised neither to 
 accept them nor to reject them without careful examination. They are 
 good books for private reading. 
 
 GEORGE W. JONES, Pablisher, 
 No Agents. Ithaca, N. Y.