—,let unity be subtracted from each, and (——1————1, or)

that is, a-b:b::c-d:d; this is called DIVI

DENDO. Euclid 17, 5.

68. In like manner, the first is to its excess above the second, as the third to its excess above the fourth.

c-d: c, and invertendo (Art. 63.) a: a- ·b::c:c

CONVERTENDO.

-d; this is

69. Hence, because a-b: a :: c— d: c, the excess of the first above the second is to the first, as the excess of the third above the fourth to the fourth.

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

Let abcd, then will a+b: a- -b : : c+d: c−d; for

71. Hence, the difference of the first and second is to their sum, as the difference of the third and fourth to their sum

For since a+b: a−b :: c+d: c-d, therefore invertendo a-ba+b:: c−d: c+d.

72. If several quantities be proportionals, as any one of the antecedents is to its consequent, so is the sum of any number of the antecedents, to the sum of their respective consequents.

Let a b:cd::e:f :: gh: kl:: m : n, &c. then will aba+c+e+g+k+m: b+d+f+h+l+n. Because a: b :: cd, therefore ad=bc, and ab=ba; also, because a : b:: ef, therefore af=be; in like manner, ah==bg, al=bk, and an= bm: wherefore (ad+af+ah+al+an=bc+be+bg+bk+bm, or) axd+f+h+1+n=bxc+e+g+k+m, wherefore a b:: c+

:

e+g+k+m: d+f+h+l+n; and the like may be proved, whatever number of antecedents and their respective consequents be taken.

73. If four quantities be proportionals, and if equimultiples or equal parts of the first and second, and equimultiples or equal parts of the third and fourth, be taken, the resulting quantities will likewise be proportionals.

For in each case, (by multiplying extremes and means,)

74. Hence, if two quantities be prime to each other, they are the least in that proportion.

75. If four quantities be proportionals, and the first and third be multiplied or divided by any quantity, and also if the second and fourth be multiplied by the same or any other quantity, the results will be proportionals.

For in each case, (multiplying extremes and means,) ad=bc, or

76. Hence, if four quantities be proportionals, their equimultiples, as also their like parts, are proportionals.

77. Hence also, if instead of the first and second terms, or of the first and third, or of the second and fourth, or of the third and fourth, other quantities proportional to them be substituted, the results in each case will be proportionals.

78. In several ranks of proportional quantities, if the corresponding terms be multiplied together, the product will be proportionals.

aek: bfl:: cgm: dhn, and the like may be shewn of any number

of ranks.

79. Hence it follows, that the like powers of proportional quantities (viz. their squares, cubes, &c.) are proportionals.

Also a b c d, &c. then by multiplying two of these ranks together, as in the former article, we have a2 : b2 : : c2 : d2, and by multiplying all the three, a3 : b3 :: c3 : d3 ; and the like may be shewn of all higher powers whatever.

80. Hence also the like roots of proportional quantities are proportionals.

α

For, let a: b :: c: d, then will a'r : br:: cr: dr; for ==

с

d'

therefore √

a

=

b

that is, = or ar: bì :: cr: bd

d, and the same may be shewn of any other roots.

The operation described in the three foregoing articles, is called COMPOUNDING THE PROPORTIONS.

81. If there be any number of quantities, and also as many others, which taken two and two in order are proportionals, namely, the first to the second of the first rank, as the first to the second of the other rank; the second to the third of the first rank, as the second to the third of the other rank, and so on to the last quantity in each; then will the first be to the last of the first rank, as the first to the last of the other rank.

Then will a ef:l; for if the above four proportions he compounded, (Art. 78.) we shall have abcd lcde :: fghk: ghkl,

may be demonstrated of any number of ranks.

This is called EX ÆQUALI IN PROPORTIONE ORDINATA, or simply EX QUO ORDINATO. Euclid 22, 5.

82. If there be any number of quantities, and as many others, which taken two and two in cross order are proportionals, namely, the first to the second of the first rank, as the last but one to the last of the other rank; the second to the third of the first rank, as the last but two to the last but one of the other rank, and so on in cross order; then will the first be to the last of the first rank, as the first to the last of the other rank.

a: b ::k: l
b:c::h: k
c: d::g
h
d:e::f: g

Then will a e::f:l; for compounding the above four proportions, (Art. 78.) there arises abcd: bcde: khgf: Ikhg, or abcd khgf

=

f:

α f that is,) a = 1, wherefore a : e :: ƒ : 1, which was bede lkhg

e

to be shewn; and the like may be proved of any number of ranks. This is called EX EQUALI IN PROPORTIONE PERTURBATA, or simply, EX QUO PERTURBATO. Euclid 23, 5.

INVERSE, OR RECIPROCAL PROPORTION.

83. The foregoing articles treat of the properties of what is called DIRECT PROPORTION, where the first is to the second as the third is to the fourth; but when the terms are so arranged,

f It must be understood, that what we have delivered on proportion, refers to commensurable magnitudes only: it is in substance the same as the 5th book of Euclid's Elements, except that the doctrine there delivered includes both commensurable and incommensurable magnitudes; Euclid has effected this double object by means of his fifth definition, which although strictly general, has been justly complained of for its ambiguity and clumsiness.