AXIOMS. greater than the same multiple of EQUIMULTIPLES of the same, or of a less. equal magnitudes, are equal to one IV. another. That magnitude of which a multiple II. is greater than the same multiple Those magnitudes of which the same, of another, is greater than that or equal magr.itudes, are equimul- other magnitude. IIL PROP. I. THEOR. If any number of magnitudes be equimultiples of as many, each of ench; what multiple soever any one of them is of ils part, the same multiple shall all the first magnitudes be of all the other. Let any number of magnitudes AB, to F, therefore AG and CH together CD be equimultiples of as many are equal to (Ax. 2. 5.) E and F toothers E, F, each of each; whatso- gether: For the same reason, beever multiple AB is of E, the same cause GB is equal to E, and HD to multiple shall AB and CD together F; GB and HD together are equal be of E and I toether. to E and F together. Wherefore, as Because AB is the many magnitudes as are in AB equal same multiple of E that to E, so many are there in AB, CD, CD is of F, as many together equal to E and F together. magnitudes as Therefore, whatsoever multiple AB A B equal to E, so many is of E, the same multiple is AB and are there in CD equal BI CD together of E and F together. to F. Divide AB into Therefore, if any magnitudes, how magnitudes equal to E, many soever, be equimultiples of as viz. AG, GB; and CD many, each of each, whatsoever mulinto CH, HD, equal tiple any one of them is of its part, each of them to F: The the same multiple shall all the first number therefore of the D magnitudes be of all the other : ' For magnitudes CH, HD, • the same demonstration holds in any shall be equal to the number of magnitudes, which was number of the others, AG, GB : And • here applied to two.' Q. E. D. because AG is equal to E, and CH are in be PROP. II. THEOR. If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth, then shall the first together with the fijih be the same multiple of the second, that the third together with the sixth is of the fourth. Let AB the first, be the same mul- third is of F the fourth ; and BG tł. tiple of C the second, that DE the fifth, the same multiple of ctho second, that EH multiple of C, that DH is of F; that the sixth is of F DI is, AĞ the first and fifth together, is the fourth: Then A the same multiple of is AG the first, E the second C, that together with the B DH the third and A fifth, the same sixth together is of E multiple of C the the fourth F. If, B Second, that DH therefore, the first bé C KI third, together with the sixth, is of the same multiple, F the fourth. &c. Q. E. D. Because AB is the same multiple H' of C, that DE is of F; there are as COR. From this many magnitudes in AB equal to C, “it ie plain, that if as there are in DE equal to F: In • any number of magnitudes AB, BG, like manner, as many as there are in • GH, be multiples of another C; and BG equal to C, so many are there in as many DE, EK, KL be the same EH equal to F: As many, then, as ' multiples of F, each of eacti; the are in the whole AG equal to C, 80 ' whole of the first, viz. AH, is the many are there in the whole DH samé multiple of C, that the whole equal to F; therefore AG is the same • of the last, viz. DL, is of F.' 6 PROP. III. THEOR. If the first be the same multiple of the second, which the third is of the fourth ; and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth. Let A the first, be the same mul. therefore of the magnitudes EK, KF, tiple of B the second, that the third shall be equal to the number of the is of D the fourth; and of A, C let others GL, LH: And because A is the equimultiples ÉF, GH be taken: the same multiple of B, that C is of Then EF is the same multiple of B, D, and that EK is equal to A, and that GH is of D. GL to C; therefore EK is the same Because EF is the same multiple multiple of B, that GL is of D: For of A, that GH is of C, there are as the same reason, KF is the same many magnitudes in ÉF equal to A, multiple of B, that LH is of D; and so, if there be more parts in EF, GH PI equal to A, C: Because, therefore, H the first EK is the same multiple of the second B, which the third GL is KI L of the fourth D, and that the fifth KF is the same multiple of the second B, which the sixth LH is of the fourth D; EF the first, together with the G C fifth, is the same multiple (2. 6.) of the second B, which GH the third, as are in GH equal to C: Let EP be together with the sixth, is of the divided into the magnitudes EK, KF, fourth D. If, therefore, the first, &c. each equal to A, and GH into GL, &c. Q. E. D. LH, each equal to C: The number El A PROP. IV. THEOR. If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ralio to any equimultiples of the second and fourth, vis. the equimultiple of the first shall have the * same ralio to that of the second, which the cquimultiple of the third • has lo that of the fourth.' Let A the first, have to B the se- have been taken certain equimultiples cond, the same ratio which the third M, N; if therefore K be greater than C has to the fourth D; and of A and M, L is greater than N: and if equal, C let there be taken any equimultiples equal ; if less, less. (5. Def. 5.) And K, whatever E, F; and of B and D any L, are any equimultiples whatever of equimultiples whatever G, H: Then E, F; and M, N any, whaterer of E has the same ratio to G, which F G, H: As therefore E is to G, so is has to H. (5. Def. 5.) F to H. Therefore, if Take of E and F any equimultiples the first, &c. Q. E. D. whatever K, L, and of G, H, any equimultiples whatever M, N : Then, Cor. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth : And in like manner, the first and the third have the same ratio to any equimultiples what. ever of the second and fourth. Let A the first, have to B the se. K E A B GM cond, the same ratio which the third C has to the fourth D, and of A and L F C DH N Ν C let E and F be any equimultiples whatever; then E is to B, as F to D. Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as before, that K is the same multiple of A, that L is of C: And because A is to B, as C is to D, and of A and C certain because E is the same multiple of A, equimultiples have been taken, viz. that F is of C; and of E and F have K and L; and of B and D certain been taken equimultiples K, L; there- equimultiples G, H; therefore, if K fore K is the same multiple of A, that be greater than G, L is greater than L is of C; (3. 6.) For the same rea- H; and if equal, eqnal; if less, less : son, M is the same multiple of B, (5. Def. 5.) And K, L are any equithat N is of D: And because, as A multiples of E, F, and G, H any is to B, so is C to D, (Hypoth.) and whatever of B, D: as therefore E is of A and C have been taken certain to B, so is F to D: and in the same equimultiples K, L; and of B and Dway the other casc is demonstrated. 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 711- G 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 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. FB shall be the same I Wherefore, since AE is the same mul. multiple of the re 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 equimultiples 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. Ax. 5.) Take away the be equimultiples of the two E, F, common magnitude CH, then the reand ÁG, 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 equal to E; HD is same multiple of C. equal to F: Make F: Make CK the CK equal to F; and same multiple of G because AG is the G H F, that GB is of same multiple of E, | E: And because . that CH is of F, and B D E F AG is the same BD HF 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 E that 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 E, that CD is of F; therefore ÅH is the same multiple of F, that CD is of that KC is of F, and that KC is equal it: wherefore KH is equal to CD: to HD; therefore HD is the same 1. Az. 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 ; Dany 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 B Ein 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 |