Posts tagged 'multiple'

G.170 multiple of a magnitude

The greater is a:term:**multiple** <multiple> of the less when it is:term:measured <part-aliquot> by the less.

G.171 there number magnitudes

If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.

G.172 magnitude multiple fifth

If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth is the same multiple of the second that the sum of the third and sixth is of the fourth.

G.175 magnitude multiple equimultiples

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also :term:`ex aequali <G.174>` the magnitudes taken is equimultiples respectively, the one of the second and the other of the fourth.

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.178 magnitude multiple magnitude

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder is the same multiple of the remainder that the whole is of the whole.

G.179 magnitudes equimultiples magnitudes

If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.

G.183 equal magnitudes ratio

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

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.189 ratios ratio

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

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.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.193 parts ratio multiples

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

G.195 magnitudes proportional proportional

If four magnitudes be proportional, they is proportional alternately.

G.198 magnitudes proportional componendo

If magnitudes be proportional componendo, they is proportional separando.

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.210 triangles parallelograms height

Triangles and parallelograms which are under the same height are to one another as their bases.

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.256 multiple of a number

The greater:term:number <G.248> is a:term:**multiple** <multiple> of the less when it is:term:measured <G.253> by the less.

G.257 number part number

If a number be a part of a number, and another be the same part of another, the sum is the same part of the sum that the one is of the one.

G.263 proportional numbers

Numbers are:term:**proportional** <G.180> when the first is the:term:same multiple <G.256>, or the:term:same part <G.253>, or the:term:same parts <G.254>, of the second that the third is of the fourth.

G.371 unequal magnitudes set

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there is left some magnitude which is less than the lesser magnitude set out.

G.531 parallelepipedal solid cut

If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.

G.571 cylinder cut plane

If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.

G.600 construct a dodecahedron and comprehend it in a

Construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures, and to prove that the side of the dodecahedron is the irrational segment called apotome.