Elements of Algebra

128 ELEMENTS SECT. III. 384. But if the difference and the less of the two numbers, that is, if $d$ and $b$ were given, we might determine the greater number by adding together the difference and the less number, which gives $a = b+d$; for if we take from $b+d$ the less number $b$, there remains $d$, which is the known difference: suppose, for example, the less number is 12, and the difference 8, then the greater number will be 20. 385. Lastly, if beside the difference $d$, the greater number $a$ be given, the other number $b$ is found by subtracting the difference from the greater number, which gives $b = a-d$; for if the number $a-d$ be taken from the greater number $a$, there remains $d$, which is the given difference. 386. The connexion, therefore, among the numbers, $a$, $b$, $d$, is of such a nature as to give the three following results: 1st. $d = a-b$; 2d. $a = b+d$; 3d. $b = a-d$; and if one of these three comparisons be just, the others must necessarily be so also; therefore, generally, if $z = x+y$, it necessarily follows, that $y = z-x$, and $x = z-y$. 387. With regard to these arithmetical ratios we must remark, that if we add to the two numbers $a$ and $b$, any number $c$, assumed at pleasure, or subtract it from them, the difference remains the same; that is, if $d$ is the difference between $a$ and $b$, that number $d$ will also be the difference between $a+c$ and $b+c$, and between $a-c$ and $b-c$. Thus, for example, the difference between the numbers 20 and 12 being 8, that difference will remain the same, whatever number we add to, or subtract from, the numbers 20 and 12. 388. The proof of this is evident: for if $a-b=d$, we have also $(a+c)-(b+c)=d$; and likewise $(a-c)-(b-c)=d$. 389. And if we double the two numbers $a$ and $b$, the difference will also become double; thus, when $a-b=d$, we shall have $2a-2b=2d$; and, generally, $na-nb=nd$, whatever value we give to $n$. X because suppose the extremes any two numbers whatever viz $a$ & $q$ then by the nature of an arith-metical proportion ($a-b=p-q$)