The Elements of Euclid; viz. the first six books, together with the eleventh and twelfth. Also the book of Euclid's Data. By R. Simson. To which is added, A treatise on the construction of the trigonometrical canon [by J. Christison] and A concise account of logarithms [by A. Robertson].

fore EF and FG are each of them greater than D: and in Book V. every one of the cases, take H the double of D, K its triple, and so on, till the multiple of D be that which first becomes greater than FG: Let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

Then, because L is the multiple of D, which is the first that becomes greater than FG, the next preceding multiple K is not greater than FG; that is, FG is not less than K: And since EF is the same multiple of AC, that FG is of CB; FG is the same multiple of CB, that EG is of AB; wherefore EG and FG are equimultiples of AB and 1. 5. CB And it was shown, that FG was

Fig. 2.

Fig. 3.
E

than L; but FG is not

G B

greater than L; and

EG, FG are equi- L K H D

multiples of AB, BC,

and L is a multiple

L K D

7 Def. $.

of D; therefore b AB has to D a greater ratio than BC has to D. Also D has to BC a greater ratio than it has to AB: For, having made the

same construction, it may be shown, in like manner, that
Lis greater than FG, but that it is not greater than EG: and
Lis a multiple of D; and FG, EG are equimultiples of CB,
AB; therefore D has to CB a greater ratio than it has to
AB. Wherefore, of unequal magnitudes, &c. Q. E. D.

BOOK V.

PROP. IX. THEOR.

See N. 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 same ratio to C; A is equal to B. For, if they are not equal, one of them is greater than the other: Let A be the greater; then, by what was shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C, Let such multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F: But, because A is to Cas B is to C, and of A, B, are taken equimultiples D, E, and of C is taken a multiple F; and that D is greater than 5 Def. 5. F; E shall also be greater than Fa; but E is not greater than F; which is im- A possible; A therefore and B are not unequal; that is, they are equal.

Next, let C have the same ratio to each
of the magnitudes A and B; A is equal
to B: For, if they are not, one of them B
is greater than the other; let A be the
greater; therefore, as was shown in Prop.

8th, there is some multiple F of C, and some equimultiples E and D, of B and A such, that F is greater than E, and not greater than D; but because C is to B, as C is to A, and that F, the multiple of the first, is greater than E, the multiple of the second; F the multiple of the third, is greater than D, the multiple of the fourtha: But F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q. E. D.

BOOK V.

PROP. X. THEOR.

THAT magnitude which has a greater ratio than See N. another has unto the same magnitude, is the greater of the two: And that magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two.

Let A have to C a greater ratio than B has to C; A is greater than B: For, because A has a greater ratio to C, than B has to C, there are some equimultiples of A and a 7 Def. 5. B, and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it: Let them be taken, and let D, E be equimultiples of A, B, and F a multiple of C such, that D is greater than F, but E is not greater than F: Therefore D is greater than E: And, because D and E are equimultiples of A and B, and D is greater than E; therefore A is greater than B..

Next, let C have a greater ratio to B than it has to A; B is less than A: For B there is some multiple F of C, and some equimultiples E and D of B and A such, that F is greater than E, but is not great

b4 Ax. 5.

er than D: E therefore is less than D; and because E and D are equimultiples of B and A, therefore B is bless than A. The magnitude, therefore, &c. Q. E. D.

PROP. XI. THEOR.

RATIOS that are the same to the same ratio, are the same to one another.

Let A be to B as C is to D; and as C to D, so let E be to F; A is to B, as E to F.

Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equimultiples whatever L, M, N. Therefore, since A is to B, as C to D, and G, H are taken equinultiples of A, C, and L, M, of B, D; if G be greater

#5 Def. 5,

Book V. than L, H is greater than M; and if equal, equal; and if less, less. Again, because C is to D, as E is to F, and H, K are taken equimultiples of C, E; and M, N, of D, F; if H be greater than M, K is greater than N; and if equal, equal; and if less, less: But if G be greater than L, it has

been shown that H is greater than M; and if equal, equal; and if less, less; therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less: And G, K are any equimultiples whatever of A, E; and L, N any whatever of B, F: Therefore, as A is to B, so is E to Fa. Wherefore, ratios that, &c. Q. E. D.

PROP. XII. THEOR.

If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents.

Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, so C to D, and E to F: As A is to B, so shall A, C, E together be to B, D, F together. Take of A, C, E any equimultiples whatever G, H, K;

and of B, D, F any equimultiples whatever L, M, N: Then, because A is to B, as C is to D, and as E to F; and that G, H,

K are equimultiples of A, C, E, and L, M, N, equimul- BOOK V. tiples of B, D, F; if G be greater than L, H is greater

than M, and K greater than N; and if equal, equal; and if less, lessa. Wherefore, if G be greater than L, then G, H, K * 5 Def. 5. together are greater than L, M, N together: and if equal, equal; and if less, less. And G, and G, H, K, together are any equimultiples of A, and A, C, E together; because if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the wholeb: For 1. 5. the same reason L, and L, M, N are any equimultiples of B, and B, D, F: As therefore A is to B, so are A, C, E, together to B, D, F together. Wherefore if any number, &c. Q. E. D.

PROP. XIII. THEOR.

Ir the first has to the second the same ratio which See N. the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first shall also have to the second a greater ratio than the fifth has to the sixth.

Let A the first, have the same ratio to B the second, which C the third, has to D the fourth, but C the third, to D the fourth, a greater ratio than E the fifth, to F the sixth: Also the first A shall have to the second B a greater ratio than the fifth E to the sixth F.

Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and some of D and F such, that the multiple of C is greater than the multiple of D, but the multiple of B is not greater than the multiple

of Fa: Let such be taken, and of C, E, let G, H be equi- 7 Def. 5. multiples, and K, L equimultiples of D, F, so that G be greater than K, but H not greater than L; and whatever multiple G is of C, take M the same multiple of A; and whatever multiple K is of D, take N the same multiple of

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The Elements of Euclid; viz. the first six books, together with the eleventh ...