Elements of Algebra

154 ELEMENTS SECT. III. also make $600$; consequently, by dividing $600$ by $24$ the quotient $25$ will be the fourth term required, and the whole proportion will be $24 : 15 :: 40 : 25$. In general, therefore, if the three first terms are $a : b :: c$; we put $d$ for the unknown fourth letter; and since $ad = bc$, we divide both sides by $a$, and have $d = \frac{bc}{a}$; so that the fourth term is $\frac{bc}{a}$, which is found by multiplying the second term by the third, and dividing that product by the first. 471. This is the foundation of the celebrated *Rule of Three* in Arithmetic; for in that rule we suppose three numbers given, and seek a fourth, in geometrical proportion with those three; so that the first may be to the second, as the third is to the fourth. 472. But here it will be necessary to pay attention to some particular circumstances. First, if in two proportions the first and the third terms are the same, as in $a : b :: c : d$, and $a : f :: c : g$, then the two second and the two fourth terms will also be in geometrical proportion, so that $b : d :: f : g$; for the first proportion being transformed into this, $a : c :: b : d$, and the second into this, $a : c :: f : g$, it follows that the relations $b : d$ and $f : g$ are equal, since each of them is equal to the relation $a : c$. Thus, for example, if $5 : 100 :: 2 : 40$, and $5 : 15 :: 2 : 6$, we must have $100 : 40 :: 15 : 6$. 473. But if the two proportions are such, that the mean terms are the same in both, I say that the first terms will be in an inverse proportion to the fourth terms: that is, if $a : b :: c : d$, and $f : b :: c : g$, it follows that $a : f :: g : d$. Let the proportions be, for example, $24 : 8 :: 9 : 3$, and $6 : 8 :: 9 : 12$, we have $24 : 6 :: 12 : 3$; the reason is evident; for the first proportion gives $ad = bc$; and the second gives $fg = bc$; therefore $ad = fg$, and $a : f :: g : d$, or $a : g :: f : d$. 474. Two proportions being given, we may always produce a new one by separately multiplying the first term of the one by the first term of the other, the second by the second, and so on with respect to the other terms. Thus, the proportions $a : b :: c : d$, and $e : f :: g : h$ will furnish this, $ae : bf :: cg : dh$; for the first giving $ad = bc$, and the second giving $eh = fg$, we have also $adeh = bcfg$; but now $adeh$ is the product of the extremes, and $bcfg$ is the product of the means in the new proportion: so that the two products being equal, the proportion is true.