be equal to 5B. Then C fhall be equal to 5D. Take any Number. For Example, 2, by which let 5 be multiplied, and the Product will

A: B:: C: D

be 10: And let 2A, 2C, be Equimultiples of the first and third Magnitudes A and C: 2A, 10B, 2C, 10D Alfo, let 10B and 10D be Equimultiples of the fecond and fourth Magnitudes B and D. Then (by Def. 5.) if 2A be equal to 10B, 2C fhall be equal to toD. But fince A (from the Hypothefis) is five Times B, 2A fhall be equal to 10B; and fo 2C equal to 10D, and C equal to 5D; that is, C will be five Times D. W. W. D.

Secondly, Let A be any Part of B; then C will be the fame Part of D. For because A is to B, as C is to D; and fince A is fome Part of B; then B will be a Multiple of A: And fo (by Cafe 1.) D will be the fame Multiple of C, and accordingly C fhall be the fame Part of the Magnitude D, as A is of B. W.W.D.

Thirdly, Let A be equal to any Number of whatfoever Parts of B. I fay, C is equal to the fame Number of the like Parts of D. For Example: Let A be a fourth Part of five Times B; that is, let A be equal to B. I fay, C is alfo equal to D. For because A is equal to B, each of them being multiplied by 4, then 4A will be equal to 5B. And fo if the Equimultiples of the first

and third, viz. 4A, 4C be af- A: B:: C: D. fumed; as alfo the Equimultiples 4A, 5B, 4C, 5D. of the second and fourth, viz. 4A, 5B, 4C, 5D. 5B, 5D, and (by the Definition) if 4A is equal to 5B; then 4C is equal to 5D. But 4A has been proved equal to 5B, and fo 4C fhall be equal to 5D, and Cequal to D. W.W.D.

n

And univerfally, if A be equal to B, C will be

equal to D, For let A and C

m

m

be multiplied by m, and B and A: B:: C: D

D by n. And because A is equal

to-B; mA fhall be equal to

m

mA, nB, mC, D

nB; wherefore (by Def. 5.) mC will be equal to »D,

VII. When of Equimultiples, the Multiple of the firft exceeds the Multiple of the fecond, but the Multiple of the third does not exceed the Multiple of the fourth; then the first to the fecond is faid to have a greater Proportion than the third to the fourth.

VIII. Analogy is a Similitude of Proportions. IX. Analogy at least confifts of three Terms. X. When three Magnitudes are Proportionals, the firft is faid to have to the third, a Duplicate Ratio to what it has to the fecond.

XI. But when four Magnitudes are Proportionals, the first hall have a triplicate Ratio to the fourth of what it has to the fecond; and so always one more in Order, as the Proportionals fhall be extended.

XII. Homologous Magnitudes, or Magnitudes of a like Ratio, are faid to be fuch whofe Antecedents are to the Antecedents, and Confequents to the Confequents.

XIII. Alternate Ratio, is the comparing of the An tecedent with the Antecedent, and the Confequent with the Confequent.

XIV Inverse Ratio, is when the Confequent is taken as the Antecedent, and fo compared with the Antecedent as a Confequent.

XV. Compounded Ratio, is when the Antecedent and Confequent taken both as one, is compared to the Confequent itself.

XVI. Divided Ratio, is when the Excess wherein the Antecedent exceeds the Confequent, is compared with the Confequent.

XVII. Converse Ratio, is when the Antecedent is compared with the Excefs, by which the Antece dent exceeds the Confequent.

XVIII. Ratio of Equality, is where there are taken more than two Magnitudes in one Order, and a like Number of Magnitudes in another Order, comparing two to two being in the fame Pro

portion;

i

PROPOSITION IV.

THEOREM.

If the firft have the fame Proportion to the fecond as the third to the fourth; then alfo fhall the Equimultiples of the first and third have the fame Proportion to the Equimultiples of the fecond and fourth, according to any Multiplication whatsoever, if they be fo taken as to answer each other.

ET the firft A have the fame Proportion to the fecond B as the third C hath to the fourth Di and let E and F, the Equimultiples of A and C, be any how taken; as alfo G, H, the Equimultiples of B and D. I fay E is to Gas F is to H.

For take K and L, any Equimultiples of E and F; and alfo M and N of G and H.

Then because E is the fame Multiple of A, as F

is of C, and K, L are

taken Equimultiples of E, K EA B G M
F, K will be the fame

Multiple of A, as L is of L F C D H N
C. For the fame Reason,
M is the fame Multiple of
B, as N is of D. And fince
A is to B, as C is to D,
and K and L are Equimul-
tiples of A and C; and alfo
M and N Equimultiples of
B and D. If K exceeds
M, then + L will exceed N;
if equal, equal; or less, less.
And K, Lare Equimultiples
of E, F, and M, N, any
other Equimultiples of GH,
Therefore, as E is to G, fe

hall

Def. 5.

fhall F be to H. Wherefore, if the first have the
‡
fame Proportion to the fecond, as the third to the
fourth; then alfo fhall the Equimultiples of the first
and third have the fame Proportion to the Equimultiples
of the fecond and fourth, according to any Multiplication
whatsoever, if they be fo taken as to answer each;
which was to be demonftrated,

Because it is demonstrated, if K exceeds M, then L will exceed N; and if it be equal to it, it will be equal; and if lefs, leffer. It is manifeft likewife, if M exceeds K, that N fhall exceed L; if equal, equal; but if lefs, lefs. And therefore as G is to E, *Def. 5. fo is * H to F..

Coroll. From hence it is manifeft, if four Magnitudes be proportional, that they will be alfo inversely proportional.

PROPOSITION V.

THEOREM.

If one Magnitude be the fame Multiple of another Magnitude, as a Part taken from the one is of a Part taken from the other then the Refidue of the one fhall be the fame Multiple of the Refidue of the other, as the whole is of the whale.

ET the Magnitude AB be the fame Multiple
of the Magnitude CD, as the Part taken away
AE is of the Part taken away CF. I
fay that the Refidue EB is the fame
Multiple of the Refidue FD, as the
whole AB is of the whole CD.

For let EB be fuch a Multiple of
CG as AE is of CF.

B

G

C

E

F

A D

Then because AE is the fame Multiple of CF, as E B is of CG, AE of this will be the fame Multiple of CF, as AB is of GF. But AE and AB are put Equimultiples of CF and CD. Therefore A B is the fame Multiple of GF as of CD; and fo GF is † equal to CD. Now let CF, which is common, be taken away; and the Refidue

+2 Axim

of this.

GC

GC is equal to the Refidue DF. And then be-
caufe AE is the fame Multiple of CF, as EB is
of CG, and CG is equal to DF; AE fhall be
the fame Multiple of CF, as EB is of FD. But
AE is put the fame Multiple of CF as AB is of CD. .
Therefore EB is the fame Multiple of FD, as
AB is of CD: and fo the Refidue EB is the fame
Multiple of the Refidue. FD, as the whole A B is of
the whole CD. Wherefore, if one Magnitude be the
Jame Multiple of another Magnitude, as a Part taken
from the one is of a Part taken from the other; then
the Refidue of the one shall be the fame Multiple of the
Refidue of the other, as the whole is of the whole;
which was to be demonftrated.

PROPOSITION VI.

THEOREM.

If two Magnitudes be Equimultiples of two Magnitudes, and fome Magnitudes Equimultiples of the fame be taken away then the Refidues are either equal to thofe Magnitudes, or elfe Equimultiples of them.

ET two Magnitudes AB, CD, be Equimulti

nitudes AG, CH, Equimultiples of the fame E, F, be taken from AB, CD. I fay, the Refidues GB, HD, are either equal to E, F, or are Equimultiples

of them.

A

For firft, Let GB be equal to E. I fay, HD is alfo equal to F. For let CK be equal to F. Then because A G is the fame Multiple of E, as CH is of F; and GB is equal to E; and CK to F; AB will be* the fame Multiple of E, as KH is of F. But AB and CD are put Equimultiples of E and F. Therefore KH is the fame Multiple of F, as CD is of F.

And because KH and CD are

G

K

C

* 1 of this.

H|

B D E F

Equimultiples of F; KH will be equal to CD.