PROP. V. THEOR. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; The remainder shall be the same multiple of the remainder, that the whole is of the whole. Let the magnitude that AB is of CD; Therefore EG is AB be the same - GI the same multiple of CD that AB is Itiple of CD, that AE of CD; wherefore EG is equal to taken from the first, is A AB. (1. Ax. 5.) Take from them the of CF taken from the common magnitude AE; the remainother; the remainder der AG is equal to the remainder EB. EB shall be the same E Wherefore, since AE is the same mul. multiple of the re F tiple of CF, that AG is of FD, and mainder FD, that the that AG is equal to EB; therefore whole AB is of the B D AE is the same multiple of CF, that whole CD. EB is of FD: But AE is the same Take AG the same multiple of FD, multiple of CF, that AB is of CD; that AE is of CF: therefore AE is therefore EB is the same multiple of (1. 5.) the same multiple of CF, that FD, that AB is of CD. Therefore, ÈG is of CD: But AE, by the hypo- if one magnitude, &c. Q. E. D. thesis, is the same multiple of CF, PROP. VI. THEOR. If two magnitudes be equimultiples of two others, and if equimoltiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples of them. Let the two magnitudes AB, CD equal CD: (1. As. 5.) Take away the be equimultiples of the two E, F, common magnitude CH, then the reand AG, CH taken from the first two mainder KC is equal to the remainbe equimultiples of the same E, F; der HD: But KC is equal to F; HD the remainders GB, HD are either therefore is equal to F. equal to E, F, or equimultiples of But let GB be them. a multiple of E; K First, Let GB be then HD is the A equal to E; HD is same multiple of equal to F: Make F: Make CK the CK equal to F; and same multiple of GI FL because AG is the F, that GB is of same multiple of E, E: And because that CH is of F, and B D F AG is the same B that GB is equal to multiple of E, E, and CK to F; therefore AB is the that CH is of F; same multiple of E, that KH is of F. and GB the same multiple of E, that But AB, by the hypothesis , is the CK is of F; therefore AB is the same same multiple of Ethat CD is of F; multiple of E, that KH is of F: (2. therefore KH is the same multiple of 5.) But AB is the same multiple of F, that CD is of F; wherefore KH is Ě, that CD is of F; therefore KH is G H the same multiple of F, that CD is of that KC is of F, and that KC is equal it: wherefore ki is equal to CD:' to HD; therefore HD is the same (1. As, 5.). Take away cH from inultiple of F, that GB is of E: If both; therefore the remainder KC is therefore two magnitudes, &c. Q. equal to the remainder HD: And be- E. D. cause GB is the same multiple of E, PROP. A, THEOR. If the first of four magnitudes has to the second the same ratio which the third has to the fourth ; then, if the first be greater than the second, the third is also greater than the fourth ; and if equal, equal ; if less, less. Take any equimultiples of each of wherefore also the double of the them, as the doubles of each ; then, third is greater than the double of the by Def. 5th of this book, if the double fourth; therefore the third is greatof the first be greater than the double er than the fourth : In like manner, of the second, the double of the third is if the first be equal to the second, greater than the double of the fourth; or less than it, the third can be provbut, if the first be greater than the ed to be equal to the fourth, or second, the double of the first is less than it. Therefore, if the first, greater than the double of the second; &c. Q. E. D. PROP. B. THEOR. If four magnitudes are proportionals, they are proportionals also when taken inversely. If the magnitude A be to B, as C as C is to D, and of A and C, the first is to D, then also inversely B is to A, and third, G and H are equimultiples; as D to C. and of B and D, the second and Take of B and fourth, E and F are equimultiples; D any equimul and that G is less than E, H is also tiples whatever (5. Def. 3) less than F; that is, F is E and F; and greater than H; if therefore E be of A and C any greater than G, F is greater than H: equimultiples în like manner, if E be equal to G, whatever G and F may be shewn to be equal to H: H. First, let E and, if less, less ; and E, F, are any be greater than equimultiples whatever of B and D, G, then G is less and G, H any whatever of A and C; than E; and be. therefore, as B is to A, so is D to C. If, cause A is to B, then, four magnitudes, &c. Q. E. D. PROP. C. THEOR. If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second, as the third is to the fourth. Let the first A be :he same mule is of the fourth D: A is to B as C is tiple of B the second, that C the third to D Take of A and of D; that is, if E be greater than G, C any equirul F is greater than H: In like manner, tiples whatever if E be equal to G, or less, F is equal E and F; and of to H, or less than it. But E, F are B & Dany equie equimultiples, any whatever, of A, C, multiples' what and G, H, any equimultiples whatever G and H: ever of B, D. Therefore A is to B, Then, because A E G F H as C is to D. (5. Def. 5.) is the same multiple of B that C Next, let the is of D; and that first A be the E is the same same part of the multiple of A, second B, that that F is of C; the third C is of E is the same the fourth D: A multiple of B, is to B, as C is, that F is of D'; (3. 5.) therefore E to D : For B is the same multiple of and F are the same multiples of B A, that D is of C: wherefore, by the and D: But G and H are equimul- preceeding case, B is to A, as D is to tiples of B and D; therefore, if E be C; and inversely (B. 5.) A is to B as a greater multiple of B than G is, F C is to D. Therefore, if the first be is a greater multiple of D than H is the same multiple, &c. Q. E. D. PROP. D. THEOR. If the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth, Let A be to B, as C is to D; and But A is equal to E, therefore C is first let A be a multiple of B; C is equal to F: (A. 5.) And F is the the same multiple of D. same multiple of D, that A is of B. Take E equal to A, and whatever Wherefore C is the same multiple of multiple A or E D, that A is of B. is of B, make F Next, let the first A be a part of the same mula the second B; C the third is the same tiple of D: Then, part of the fourth D. because A is to : Because A is to B, as C is to D; B, as C is to D; then, inversely, B is (B. 5.) to A, as and of B the se А в с D D to C: But A is a part of B, therecond, and D the E fore B is a multiple of A; and, by the fourth equimul preceding case, D is the same multiples have been tiple of C; that is, C is the same part taken E and F; of D, that A is of B: Therefore, if the A is to E, as c first, &c. Q. E. D. to F: (Cor. 4.5.) R PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magnitude ; and the same has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and if equal, equal ; if less, less : And and C any other. A and B have each D, E are any equimultiples of A, B, of them the same ratio to C, and C and F is any multiple of C. Therehas the same ratio to each of the mag. fore, (5. Def. 5.) as A is to C, so is B nitudes A and B. to C. Take of A and B any equimultiples Likewise C has the same ratio to whatever D and A, that it has to B: For, having made E, and of C any the same construction, D may in like multiple wbai manner be shewn equal to E: Thereever F: Then, fore, if F be greater than D, it is likebecause D is the wise greater than E; and if equal, same multiple of D A equal; if Jess, less : And F is any A, that E is of multiple whatever of C, and D, É B, and that A is E B are any equimultiples whatever of equal to B ; D is A, B. Therefore C is to A, as C (i. Ax. 5.) equal is to B. (5. Def. 5.) Therefore equal to E: Therefore, magnitudes, &c. Q. E. D. if D be greater than F, E is greater than F; PROP. VIII. THEOR. Of unequal magnitudes, the greater has the greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater. a Let AB, BC be unequal magni- in Fig. 2. and 3.) this magnitude can tudes, of v:hich AB is the greater, be multiplied, so as to become greater and let D be any magnitude wható than D, whether it be AC, or CB. ever: AB has a Fig. 1. Let it be multiplied, until it become greater ratio to D greater than D, and let the other be than BC to D: E multiplied as often ; and let EF be And D has the multiple thus taken of AC, and greater ratio to FI А FG the same multiple of CB : ThereBC than into AB. fore EF and FG are each of them If the magni C greater than D: And in every one of tude which is not GB the cases, take H the double of D, K the greater of the L K HD its triple, and so on, till the multiple two AC, CB be of d be that which first becomes not less than D, greater than FG: Let L be that mula take EF, FG, the tiple of D which is first greater than toubles of AC, PG, and K the multiple of D which is CB, as in Fig. 1. next less than L. But if that which is not the greater Then, because Lis the multiple of D, of the two AC, CB be less than D (as which is the first that becomes greater than FG, the next preceding multiple Also'D has to BC agreater ratio than K is not greater than FG ; that is, FG it has to AB: For, having made the is not less than K: And since EF is the same construction, it may be shewn, same muluple of AC, that FG is of in like manner, that L is greater than ÇB; FG is the same multiple of CB FG, but that that EG is AB; (1. 5.) wherefore it is not greatEG and FG are equimultiples of AB er than EG; and CB: And it was shewn, that FG and I is a F was not less than K, and, by the con- multiple of A struction, EF is greater than D; D; and FG, therefore the whole EG is greater EG are than K and D together: But K, to- quimultiples gether with D, is equal to L; there- of CB, AB; | LKD fore EG is greater than L; but FG therefore D is not greater than L; and EG, FG has to CB a are eqnimultiples of AB, BC, and L greater is a multiple of D; therefore (7. Def. tio (7. Def. 5.) AB has to D a greater ratio than BC .) than it has to AB. Wherefore, has to D. of unequal magnitudes, &c. Q. E. D. Fc e I KIDGB ra PROP. IX. THEOR. was Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. Let A, B have each of them the that D is greater than F; E shall alsame ratio to C: A is equal to B: so be greater than F; (5. Def. 5.) For, if they are but E is not greater than F, which is not equal, one of impossible; À therefore and B are not them is greater unequal ; that is, they are equal. than the otheria a D Next, let C have the same ratio to A be each of the magnitudes A and B : A greater ; then, F o hy what is equal to B : For, if they are not, one of them is greater than the other ; shown in the let A be the greater; therefore, as was preceding propo BA shown in Prop. 8th, there is some sition, there are multiple F of C, and soine equimulsome equimul tiples E and D, of B and A such, that tiples of A and B, and some multiple F is greater than E, and not greater of C such, that the multiple of A is than D; but because C is to B, as c greater than the multiple of C, but is to A, and that F, the multiple of the multiple of B is not greater than the first, is greater than E, the multhat of C. Let such multiples be tiple of the second; F, the multiple of taken, and let D, E, be the equimul- the third, is greater than D, the multiples of A, B, and F the multiple of tiple of the fourth : (5. Def. 5.) But C, so that D may be greater than F, Fis not greater than D, which is imand E not greater than F: But, be- possible. Therefore, A is equal to B. cause A is to C, as B is to C, and of Wherefore, magnitudes which, &c. A, B are taken equimultiples D, E, Q. E. D. and of C is taken a multiple F; and |