If A For AD

=

B C D, then (by alternation) A: C:: B: D.
BC (26),.. A: C:: B: D (33).

37. If four quantities be proportional, the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth.

Let A: B::C: D, then (by composition)

A+B B::C+D:D.

:

38. If four quantities be proportional, the difference between the first and second is to the second, as the difference between the third and fourth is to the fourth. Let A B C : D, then (by division)

A-BB:: C

39. If four quantities be proportional, the first is to the sum or difference of the first and second, as the third is to the sum or difference of the third and fourth.

Let A B C D, then (by conversion)

:

A: A+B::C: C+ D, and A: A — B::C: C

For B: A:: D: C (35), or

- D.

B

D

=

.A: BA:: C: DC.

Cor. BA: A :: DC: C (35).

40. If four quantities be proportional, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference.

Let A: B:: C: D, then (by mixing)

A+BA-B::C+D:C-D.

:

For A+B B::C+D:D (37),
..A+B:C+D::B:D (36).

Again, ABB::C-D:D (38),
..A B:CD::B: D.

C

Hence A+B:C+D:: A-B: C-D (34)
..A+BA-B::C+D: C-D (36).

41. If several quantities be proportional, as one of the antecedents is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.

Let A: B:: C:D:: E:F:: G: H, &c. that is, let A: B:: C: D, and A: B:: E: F, &c.

=

then A: B:: A+ C + E + G: B+D+F+ H.
For AD BC (26), AF BE, AH BG; also AB BA; ..
ABAD + AF + AH ≈ BA + BC + BE + BG, or
AX (B+D+F+H) = BX (A + C+E+G) (25),..
A: BA+C+E+G:B+D+F+H (33).

=

42. If there be several ranks of proportional quantities, the products of the corresponding terms will be proportional.

If A: B:: C: D

and E: F::G: H

and I: K:: L: M

then AEI: BFK:: CGL : DHM.

For

A C E G I L

&

&

AEI CGL

B D F H K M BFK DHM'

.. AEI: BFK:: CGL: DHM.

43. If the same quantity occur in all the terms of a compound proportion, or in either of its two ratios, that quantity may be rejected, and the remaining terms will be proportional.

If A: B:: C:D

and E: A::F:C

then AE: AB:: CF: CD (42)

.. E : B :: F:D (30).

Again, let A: B:: C: D

and B: E::D:F

and E: G::F: H

then ABE: BEG:: CDF: DFH

.. A: G::C: H.

44. If four quantities be proportional, the like pow ers or roots of those quantities will be proportional.

If A: B:: C: D, then A": B" :: C": D", n being either a whole number or a fraction.

that is, A": B" : : C" : D",

45. If several quantities, A, B, C, D, E, &c. be in continued proportion, the first quantity will be to the third as the square of the first is to the square of the second; the first quantity will be to the fourth as the cube of the first is to the cube of the second, &c. that is,

A:C:: A: B, or in the duplicate ratio of A: B;
A: D::A3: B3, or in the triplicate ratio of A: B;

A: E:: A1: B4, or in the quadruplicate ratio of A: B; &c. &c.
1. Let A: B: B; C, then B'
AC (26),..
A X B9
= A x AC =

=

A x C, .. A: C:: 'A': B2 (33).

2. Let A: B::B:C::C: D, then A: C:: A: B3 (as above). But CD::A: B

.. A: D:: A3: B3 (42 & 43).

3. Let A B : B : C : ; C :D :: D: E,

then A: D:A3: B3 (as above).
But DE::A: B

.. A; E:: A+ B+.
A4 :

Scholium. The doctrine of Proportion in this tract is more general than that in Euclid's Elements, It includes the properties of both proportional numbers and of magnitudes. Euclid's Fifth Book contains the properties of proportional magnitudes only. The word quantity, employed above, denotes both numbers and magnitudes, or objects having extension. This tract is chiefly taken from the treatises of Algebra of Wood and Bridge, and is the same in substance as is taught in foreign universities, instead of the Fifth Book of Euclid.

THE MOST USEFUL PROPERTIES OF PROPORTION.

IF four quantities be proportional, the product of the extremes will be equal to the product of the means.

If the first quantity be to the second as the second to the third, the product of the extremes will be equa! to the square of the mean.

If four quantities be proportional, according as the first quantity is equal to, greater, or less than the second, the third is equal to, greater, or less than the fourth; or according as the first quantity is equal to, greater, or less than the third, the second is equal to, greater, or less than the fourth.

If the product of two quantities be equal to the product of two other quantities, these four quantities may be turned into a proportion by making the terms of one product the means, and the terms of the other product the extremes.

Quantities which have the same ratio to the same quantities are proportional.

If four quantities be proportional, they are also proportional by inversion, alternation, composition, division, conversion, and mixing.

If several quantities be proportional, as one of the antecedents is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.