BM 



MATHEMATICa PLANE GEOMETRY. 



[BOOK v. PROP. i. v. 



III. The magnitude of which a multiple or sub-multiple 

 is greater than a like multiple or sub-multiple of another, 

 it greater than that other. 



IV. Of any two magnitude* of the same kind, a mul- 

 tiple of one may be taken so great a* to exceed the other. 



Maria and Sign* tdfor Abbreviation. 



1. To exprem that four magnitudes, A, B. C, D, are 

 proportionals, they are arranged thus : A : B : : C : D ; 

 Lid this expression is read, "as A ia to B, soisCtoD ;' 

 or, " A is to B, as C is to D." 



2. And to express that magnitudes A, B, C, 1), fc, 

 *c., form a series of continual proportionals, they are 

 arranged thus: A : B : : B : C : : C : D : : D : E, tc., 

 which is read, " as A is to B, so is B to C, and C to D, 

 and D to E," <tc. 



3. Besides the marks + and for addition and sub- 

 traction, the mark </> is sometimes employed, to denote 

 the difference of the two quantities between which it is 

 placed; it is useful for this purpose when it is not stated 

 \chitk of the two quantities is the greater of the two ; 

 thus A u> B means simply the difference between A and 

 H, or rather between the things denoted by these letters, 

 without any assertion as .to which is the greater. The 

 double mark + between two quantities signifies "the 

 sum or difference" of those quantities. Tims by 6 '2 

 we should understand " 8 or 4." 



4. The terms "greater than" and "less than" being 

 of frequent use in wliat follows, convenient symbols for 

 them are introduced : when > is placed between two 

 quantities it implies that the first of them is greater than 

 the second ; and when < is placed between them, it de- 

 notes that the first of them is kss than the second. Thus 

 A > B asserts that A is greater than B, or that A exceeds 

 B ; and A < B affirms that A. is less than B, or that B 

 exceeds A. 



6. In the following propositions, magnitudes (whether 

 lines, surfaces, or solids) will be represented by the 

 capital letters, A, B, C, etc. They may, indeed, betaken 

 t<> represent any quantities whatever, whether abstract 

 or concrete ; as the reasonings will be found to apply 

 without restriction as to the nature of the things repre- 

 sented by the letters. But when multiples of these 

 quantities are taken (that is, when they are multiplied 

 by numbers), these numbers will be represented by the 

 small letters ; and, in general, by m, n, p, q. 



PROPOSITION I. THEOREM. 



If any number of magnitudes bo equimultiples of as 

 many others, each of each, whatever multiple any one 

 of the former is of the corresponding one of the latter, 

 the same multiple is the sum of all the former, and of 

 the sum of all the latter. 



First, let there be but two magnitudes mA, mB, any 

 equimultiples whatever of two others, A, B ; the sum of 

 the former shall be tin: same multiple of the sum of the 

 latter. 



For the sum of the former is m.\ + mB ; that is, m(A 

 -r- H) ; and the sum of the latter is A + B ; and m(A + 

 B) is the same multiple of A -)- B, that mA is of A, or 

 wjl! of B. 



Next, let there be three magnitudes mA, mB, mC, 

 equimultiples of the three A, B, C. 



The sum of the former three is mA + mB + mC ; 

 that is m(A + B + C), and the sum of the latter three 

 U A -f- B 4-0 ; and m(A + B +C) is the same ;i "//</;/. of 

 A -f B + C, that mA is of A, or mB of B, or mC of C. 

 Anil in the same manner is the proposition proved, when 

 there are four magnitudes equimultiples of other four ; 

 when there are five magnitudes, six magnitudes, or any 

 number of magnitudes, .'. if any number of magnitudes. 

 Ac. Q. E. D. 



PROPOSITION' II TIIKORFM. 



If, in any proportion, an antecedent and its consequent 

 be rmpeutivuly the same as an antecedent and its n 

 qut-nt in inuitlirr projMjrtion, the remaining antecedent 

 and connequont in the former, together with the remain- 



in,' antecedent Mid consequent in the latter, will form a 

 proportion. 



Let the two proportions be 



A : B : : : D .. r r\ v v 

 D f fi then \J '. It i '. B l W, 



\ : 1! : : Hi : K 



For (Def. VI.) m being any whole number whatever, C 

 is contained in mD as often as A is contained in mB, but 

 not oftener. 



In like manner E is contained in mF as often as A is 

 contained in mB, but not oftener. 



Therefore C cannot be contained oftener in mD than 

 E is contained in mF ; nor can E be contained oftener in 

 mF, than C is contained in mD ; and is any whole 

 number whatever, .'. (Def. VI.) the four magnitudes, C, 

 1), E, F, are proportionals, .'. if in any proportion, <fcc. 

 Q. E. D. 



PROPOSITION III. THEOREM. 



If, in any proportion, equimultiples of the antecedents 

 and equimultiples of the consequents be taken ; if the 

 multiple of one of the antecedents be greater than that 

 of its consequent, the multiple of the other antecedent 

 will be greater than that of its consequent. 



Let tue proportion be 



A : B : : C : D. 



If m\ > nB, then mC > nD, and conversely ; m and 

 n being any whole numbers whatever. 



For A is contained in iA exactly m times ; but, by 

 hypotheses, nB is less than niA, .'. A U contained in !' 

 less than m times. But (Dof. VI.) C is contained no 

 oftener in nD, than A is contained in nB, .'. C is con- 

 tained in nD, less than m times. 



ButC is contained in ?/iC exactly m times, .'. C is 

 contained in mC oftener than it is contained in nD, . . 

 mC nD. 



In like manner, if the hypothesis be that mC > nD, 

 may it be shown that mA > nB, .'. if in a proportion, 

 ic. Q. E. D. 



COR Since m and n may be any whole numbers 

 whatever, let each be = 1 : then it follows that : 



In a proportion, if one antecedent be greater than its 

 consequent, the other antecedent will be greater than its 

 consequent. 



PROPOSITION IV. THEOREM. 



In any proportion, according as one antecedent is 

 greater than, less than, or equal to its consequent. H 

 will the other antecedent be greater than, less than, or 

 equal to its consequent. 



It has already been proved (Prop. III., Cor.) that if 

 one antecedent be greater than if-s consequent, the other 

 antecedent will be greater than its consequent. Lot the 

 proportion be : 



A : B : : C : D. 



1st. If A < B, then C < D, and conversely. Let T? 

 A = P ; then a number m exists such that mP > A 

 (Ax. 4), .*. mB must contain A iiflencr than it contains 

 B. But mD contains C as often as mB contains A 

 (Def. VI.), .'. mD contains C oflener than mB contains 

 B ; that is, oftener than m times, .'. C < 1). Conse- 

 quently, if A < Bthen C < D. And in like manner may 

 it bo shown that if C < D then A < B. 



2nd. If A = B then C = D, and conversely. 



For when A = B, if it were possible tliat C > D, or 

 C < D, or, when C = D, if it were possible that A > B, 

 or A < B, the foregoing conclusions would bo contra- 

 dicted, .'. in any proportion, (fee. Q. E. D. 



PROPOSITION V. THEOREM 



If four magnitudes bo such, that whatever equimul- 

 tiples of the antecedents, and whatever equimultiples of 

 the consequents bo taken, the multiple of one antecedent 

 cannot be greater than that of its consequent, without 

 the multiple of the other antecedent being greater also 

 than that of its consequent, the four magnitudes are 

 prop 



Let the four magnitudes be A, B, C, D. If they are 

 not proportional, cue of the antecedents, as A, mast be 



