### Using the P/Q/Q′ scales to compute sides of right triangles

 Addition works like multiplication on C/D scales, except that when you use the right index the result appears on Q′. $$\sqrt{3^2+4^2}=5$$ Set 3 on Q opposite left index of P, read 5 on Q opposite 4 on P. $$\sqrt{20^2+99^2}=101$$ Set 99 on Q opposite right index of P, set hairline to 20 on P, read 101 under hairline on Q′. $$\sqrt{1.2^2+0.5^2}=1.3$$ Set hairline to left index of P, set 1.2 on Q′ under hairline, set hairline to 0.5 on P, read 1.3 under hairline on Q′. Subtraction works like division on C/D scales, except that when you use the right index the result moves from Q′ to Q. $$\sqrt{53^2-45^2}=28$$ Set 53 on Q opposite 45 on P, read 28 on Q opposite left index of P. $$\sqrt{119^2-60^2}=91$$ Set hairline to 60 on P, set 119 on Q′ under hairline, read 91 on Q opposite right index of P. $$\sqrt{137^2-88^2}=105$$ Set hairline to 88 on P, set 137 on Q′ under hairline, set hairline to left index of P, read 105 under hairline on Q′. $$\sqrt{117^2-108^2}=45$$ Set hairline to left index of P, set 108 on Q′ under hairline, set hairline to 117 on Q′, read 45 under hairline on P. Problems involving small values, between 1.415 and 3-ish, can be computed with improved precision by scaling. $$\sqrt{1.2^2+1.5^2}\approx 1.92094$$ Direct computation gives 1.9(3?). By scaling the arguments to 6 and 7.5 the rule gives 9605 rescaled = 1.921. $$\sqrt{23^2-15^2}\approx 17.4356$$ Direct computation gives 17.(4?). By scaling the arguments to 11.5 and 7.5 the rule gives 8715 rescaled = 17.43.

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