— GEOMETOR • elements

G.173 ratio

A:term:**ratio** <ratio> is a sort of relation in respect of size between two magnitudes of the same kind.

G.174 ex aequali (equality of ratios)

A:term:ratio <G.173>ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes; Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.

G.176 magnitudes in the same ratio

Magnitudes are said to:term:**be in the same ratio** <in-the-same-ratio>, the first to the second and the third to the fourth, when, if any:term:equimultiples <multiple> whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

G.177 magnitude ratio equimultiples

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.

G.180 proportional magnitudes

Let magnitudes which have the:term:same ratio <G.176> be called:term:**proportional** <proportional>.

G.182 invertendo (inverse ratio)

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

G.183 equal magnitudes ratio

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

G.184 magnitudes having a ratio

Magnitudes are said to:term:**have a ratio** <have-a-ratio> to one another which are capable, when multiplied, of exceeding one another.

G.185 greater ratio

When, of the:term:equimultiples <multiple>, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to:term:**have a greater ratio** <greater-ratio> to the second than the third has to the fourth.

G.186 unequal magnitudes greater

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.

G.187 magnitudes ratio equal

Magnitudes which have the same ratio to the same are equal; and magnitudes to which the same has the same ratio are equal.

G.188 magnitudes ratio greater

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

G.189 ratios ratio

Ratios which are the same with the same ratio are also the same with one another.

G.191 magnitude ratio greater

If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.

G.192 magnitude ratio greater

If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second is greater than the fourth; if equal, equal; and if less, less.

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.199 magnitudes proportional separando

If magnitudes be proportional separando, they is proportional componendo.

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.

G.202 there magnitudes others

If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if :term:`ex aequali <G.174>` the first be greater than the third, the fourth is greater than the sixth; if equal, equal; and, if less, less.

G.204 there magnitudes others

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.

G.205 there number magnitudes

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they is in the same ratio ex aequali.

G.206 there magnitudes others

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they is in the same ratio ex aequali.

G.207 magnitude ratio fifth

If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.

G.212 angle triangle bisected

If an angle of a triangle be bisected and the segment cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the segment joined from the vertex to the point of section will bisect the angle of the triangle.

G.230 duplicate ratio

When three magnitudes are:term:proportional <G.180>, the first is said to have to the third the:term:**duplicate ratio** <duplicate-ratio> of that which it has to the second.

G.231 similar triangles duplicate

Similar triangles are to one another in the duplicate ratio of the corresponding sides.

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.235 equiangular parallelograms ratio

Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

G.242 extreme and mean ratio

A segment is said to have been cut in extreme and mean ratio when, as the whole:term:line <G.2> is to the greater segment, so is the greater to the less.

G.243 extreme and mean ratio

Cut a given segment in extreme and mean ratio.

G.246 equal circles angles

In equal circles angles have the same ratio as the arcs on which they stand, whether they stand at the centres or at the arcs.

G.267 there many numbers

If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they is in the same ratio ex aequali.

G.271 number multiplying numbers

If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

G.272 numbers multiplying number

If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

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.276 least numbers ratio

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

G.289 many numbers please

Given as many numbers as we please, to find the least of those which have the same ratio with them.

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.297 find numbers in continued proportion

Find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.

G.298 many numbers please

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

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.301 plane numbers ratio

Plane numbers have to one another the ratio compounded of the ratios of their sides.

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.306 numbers prime numbers

If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.

G.309 square numbers there

Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.

G.310 triplicate ratio

When four magnitudes are < continuously >:term:proportional <G.180>, the first is said to have to the fourth the:term:**triplicate ratio** <triplicate-ratio> of that which it has to the second, and so on continually, whatever be the proportion.

G.312 cube numbers there

Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

G.313 there many numbers

If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products is proportional; and, if the original numbers by multiplying the products make certain numbers, the latter is proportional.

G.314 square measure square

If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.

G.315 cube number measure

If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.

G.319 similar plane numbers

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

G.321 similar solid numbers

Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.

G.323 mean proportional numbers

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

G.326 numbers ratio square

If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second is square.

G.327 numbers ratio cube

If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second is cube.

G.328 similar plane numbers

Similar plane numbers have to one another the ratio which a square number has to a square number.

G.329 similar solid numbers

Similar solid numbers have to one another the ratio which a cube number has to a cube number.

G.339 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two.

G.345 numbers continued proportion

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together is prime to the remaining number.

G.376 commensurable magnitudes ratio

Commensurable magnitudes have to one another the ratio which a number has to a number.

G.377 magnitudes ratio number

If two magnitudes have to one another the ratio which a number has to a number, the magnitudes is commensurable.

G.378 incommensurable magnitudes ratio

Incommensurable magnitudes have not to one another the ratio which a number has to a number.

G.379 magnitudes ratio number

If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes is incommensurable.

G.381 squares segments commensurable

The squares on segments commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on segments incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.

G.382 magnitudes proportional commensurable

If four magnitudes be proportional, and the first be commensurable with the second, the third is commensurable with the fourth; and, if the first be incommensurable with the second, the third is incommensurable with the fourth.

G.383 find two segments incommensurable

Find two segments incommensurable, the one in length only, and the other in square also, with an assigned segment.

G.384 magnitudes commensurable magnitude

Magnitudes commensurable with the same magnitude are commensurable with one another also.

G.403 find two rational segments commensurable in square only

Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment commensurable in length with the greater.

G.404 find two rational segments commensurable in square only

Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment incommensurable in length with the greater.

G.423 find the first binomial segment

Find the first binomial segment.

G.425 find the second binomial segment

Find the second binomial segment.

G.427 find the third binomial segment

Find the third binomial segment.

G.429 find the fourth binomial segment

Find the fourth binomial segment.

G.431 find the fifth binomial segment

Find the fifth binomial segment.

G.433 find the sixth binomial segment

Find the sixth binomial segment.

G.467 find the first apotome

Find the first apotome.

G.468 find the second apotome

Find the second apotome.

G.470 find the third apotome

Find the third apotome.

G.472 find the fourth apotome

Find the fourth apotome.

G.474 find the fifth apotome

Find the fifth apotome.

G.476 find the sixth apotome

Find the sixth apotome.

G.489 segment commensurable length

A segment commensurable in length with an apotome is an apotome and the same in order.

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.

G.499 square rational segment

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

G.500 area contained apotome

If an area be contained by an apotome and the binomial segment the terms of which are commensurable with the terms of the apotome and in the same ratio, the “side” of the area is rational.

G.520 segments cut parallel

If two segments be cut by parallel planes, they is cut in the same ratios.

G.543 similar parallelepipedal solids

Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.

G.547 segments proportional parallelepipedal

If four segments be proportional, the parallelepipedal solids on them which are similar and similarly described is proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the segments will themselves also be proportional.

Posts tagged 'ratio'