— GEOMETOR • elements

G.181 corresponding magnitudes

The term:term:**corresponding magnitudes** <corresponding-magnitudes> is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

G.182 invertendo (inverse ratio)

Inverse ratio means taking the:term:consequent <G.181> as antecedent in relation to the antecedent as consequent.

G.190 number magnitudes proportional

If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.

G.193 parts ratio multiples

Parts have the same ratio as the same multiples of them taken in corresponding order.

G.194 alternando (alternate ratio)

Alternate ratio means taking the:term:antecedent <G.181> in relation to the antecedent and the consequent in relation to the consequent.

G.196 componendo (composition of ratio)

Composition of a ratio means taking the:term:antecedent <G.181> together with the consequent as one in relation to the consequent by itself.

G.197 separando (separation of ratio)

Separation of a ratio means taking the excess by which the:term:antecedent <G.181> exceeds the consequent in relation to the consequent by itself.

G.200 convertendo (conversion of ratio)

Conversion of a ratio means taking the:term:antecedent <G.181> in relation to the excess by which the antecedent exceeds the consequent.

A:term:**perturbed proportion** <perturbed-proportion> arises when, there being three magnitudes and another set equal to them in multitude, as:term:antecedent <G.181> is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

G.232 similar polygons divided

Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.

G.265 there many numbers

If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.

G.268 unit measure number

If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.

G.274 least numbers ratio

The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.

G.275 numbers prime least

Numbers prime to one another are the least of those which have the same ratio with them.

G.278 numbers prime number

If two numbers be prime to any number, their product also is prime to the same.

G.285 numbers multiplying make

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

G.290 numbers find least

Given two numbers, to find the least number which they measure.

G.296 there many numbers

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

G.299 many ratios please

Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.

G.305 numbers there fall

If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.

G.323 mean proportional numbers

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

G.342 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit is measured by the same.

G.346 numbers prime number

If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

G.347 there many numbers

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the second.

G.349 numbers investigate possible

Given three numbers, to investigate when it is possible to find a fourth proportional to them.

G.368 many numbers please

If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.

G.370 many numbers please

If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product is perfect.

G.498 square rational segment

The square on a rational segment applied to the binomial segment produces as breadth an apotome the terms of which are commensurable with the terms of the binomial and moreover in the same ratio; and further the apotome so arising will have the same order as the binomial segment.

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