— GEOMETOR • elements

G.3 extremities of a segment

The extremities of a segment / A B / are points A, B.

G.4 segment

A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.

A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D / falling upon the arc from A are equal.

G.11 center of a circle

The point A within the circle ( A B ) from which all segments / A B /, / A C / to the arc are equal is called the center of the circle.

G.14 construct a segment

Construct a segment / A B / between any two points A, B.

G.17 construct an equilateral triangle

Construct an equilateral triangle on a given segment.

G.20 transfer a segment to a point

Construct at a given point a segment equal to a given segment.

G.21 cut off an equal segment

Cut off from the greater of two unequal segments a segment equal to the less.

G.24 congruent triangles by SAS

Triangles with two sides and the included angle equal are congruent (SAS).

G.25 base angles of an isosceles triangle

In an isosceles triangle, the angles at the base are equal, and if the equal sides are extended, the angles under the base are equal.

G.28 uniqueness of triangle sides on a base

On the same segment and on the same side, there cannot be constructed two distinct triangles with both pairs of conterminous sides equal.

G.29 congruent triangles by SSS

Triangles with all three corresponding sides equal have their corresponding angles equal (SSS).

G.31 bisect an angle

Bisect a given angle.

G.32 bisect a segment

Bisect a given segment.

G.34 construct a perpendicular at a point on a line

Construct a segment at right angles to a given line from a given point on it.

G.35 construct a perpendicular from a point to a line

Construct a perpendicular to a given line from a given point not on it.

G.37 adjacent angles on a line sum to two right angles

A segment standing on a line makes either two right angles or adjacent angles summing to two right angles.

G.39 supplementary adjacent angles form a line

If two segments at a point on a line on opposite sides make adjacent angles summing to two right angles, the two segments are collinear.

G.40 vertical angles are equal

Intersecting lines make their vertical angles equal.

G.41 exterior angle of a triangle is greater than interior

In any triangle, if one side is extended, the exterior angle is greater than either opposite interior angle.

G.42 two angles of a triangle are less than two right angles

In any triangle, the sum of any two angles is less than two right angles.

G.45 triangle inequality

In any triangle, the sum of any two sides is greater than the remaining side.

G.46 interior segments of a triangle

Segments constructed from the endpoints of one side of a triangle to an interior point have a lesser sum but contain a greater angle than the other two sides.

G.47 construct a triangle from three segments

Construct a triangle from three segments equal to three given segments satisfying the triangle inequality.

G.48 construct an angle equal to a given angle

Construct on a given line at a given point an angle equal to a given angle.

G.49 hinge theorem

If two triangles have two sides equal to two sides respectively, the one with the greater included angle has the greater base.

G.53 equal alternate interior angles imply parallel lines

If a transversal cutting two lines makes the alternate interior angles equal, the lines are parallel.

G.58 construct a line parallel to a given line

Construct through a given point a line parallel to a given line.

G.59 exterior angle and angle sum of a triangle

In any triangle, an exterior angle equals the sum of the two opposite interior angles, and the three interior angles sum to two right angles.

G.66 triangles on the same base in the same parallels

Triangles on the same base and in the same parallels are equal in area.

G.67 triangles on equal bases in the same parallels

Triangles on equal bases and in the same parallels are equal in area.

G.68 equal triangles on the same base are in the same parallels

Equal triangles on the same base and on the same side lie in the same parallels.

G.69 equal triangles on equal bases are in the same parallels

Equal triangles on equal bases and on the same side lie in the same parallels.

G.71 construct a parallelogram equal to a triangle

Construct, in a given angle, a parallelogram equal in area to a given triangle.

G.73 apply a parallelogram equal to a triangle on a segment

Apply to a given segment, in a given angle, a parallelogram equal in area to a given triangle.

G.74 construct a parallelogram equal to a polygon

Construct, in a given angle, a parallelogram equal in area to a given polygon.

G.75 construct a square on a segment

Construct a square on a given segment.

G.79 Pythagorean theorem

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the two legs (Pythagorean Theorem).

G.80 converse of the Pythagorean theorem

If the square on one side of a triangle equals the sum of the squares on the other two sides, the angle contained by those two sides is right.

G.82 distributive law of rectangles

If one of two segments is cut into any number of parts, the rectangle contained by the two segments equals the sum of the rectangles contained by the uncut segment and each of the parts.

G.83 square equals sum of rectangles on parts

If a segment is cut at random, the sum of the rectangles contained by the whole and each of the parts equals the square on the whole.

G.84 rectangle on whole and part

If a segment is cut at random, the rectangle contained by the whole and one of the parts equals the rectangle contained by the parts together with the square on that part.

G.85 square of a sum

If a segment is cut at random, the square on the whole equals the squares on the parts together with twice the rectangle contained by the parts.

G.87 rectangle on unequal parts plus square on difference

If a segment is cut into equal and unequal parts, the rectangle contained by the unequal parts together with the square on the segment between the points of section equals the square on the half.

G.88 rectangle on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the rectangle contained by the whole with the added segment and the added segment together with the square on the half equals the square on the segment made up of the half and the added segment.

G.89 square of a difference

If a segment is cut at random, the sum of the squares on the whole and on one of the parts equals twice the rectangle contained by the whole and that part together with the square on the remaining part.

G.90 four rectangles plus square on difference

If a segment is cut at random, four times the rectangle contained by the whole and one of the parts together with the square on the remaining part equals the square on the whole and the first part as one segment.

G.91 sum of squares on unequal parts

If a segment is cut into equal and unequal parts, the sum of the squares on the unequal parts is double the sum of the square on the half and the square on the segment between the points of section.

G.92 sum of squares on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the sum of the squares on the whole with the added segment and on the added segment is double the sum of the square on the half and the square on the half with the added segment.

G.93 cut a segment in extreme and mean ratio

Cut a given segment so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

G.94 law of cosines for obtuse triangles

In obtuse-angled triangles, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.95 law of cosines for acute triangles

In acute-angled triangles, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.97 construct a square equal to a polygon

Construct a square equal in area to a given rectilineal figure.

G.98 find the center of a circle

Find the center of a given circle.

G.99 chord of a circle lies within the circle

The segment joining any two points on the arc of a circle falls within the circle.

G.100 diameter bisects chord iff perpendicular

A segment through the center of a circle bisects a chord not through the center if and only if it cuts the chord at right angles.

G.101 intersecting chords not through the center do not bisect

In a circle, two intersecting chords that do not both pass through the center do not bisect one another.

G.102 intersecting circles have distinct centers

Two intersecting circles do not have the same center.

G.104 touching circles have distinct centers

Two circles that touch one another do not have the same center.

G.105 distances to a circle from an eccentric diameter point

From an eccentric point on the diameter of a circle, the greatest segment to the arc passes through the center, the least is the remainder of the diameter, and segments nearer the center are greater than those more remote, with at most two equal segments on opposite sides.

G.106 distances to a circle from an exterior point

From a point outside a circle, of segments falling on the concave arc the greatest passes through the center and nearer ones are greater, while on the convex arc the least lies between the point and the diameter and nearer ones are less, with at most two equal segments.

G.107 point with three equal segments to a circle is the center

A point within a circle from which more than two equal segments fall on the circle is the center of the circle.

G.108 two circles intersect in at most two points

A circle does not intersect another circle at more than two points.

G.109 line of centers of internally touching circles

If two circles touch one another internally, the line joining their centers passes through the point of contact.

G.110 line of centers of externally touching circles

If two circles touch one another externally, the segment joining their centers passes through the point of contact.

G.112 two circles touch at most at one point

A circle does not touch another circle at more than one point, whether internally or externally.

G.114 equal chords are equidistant from the center

In a circle, chords are equal if and only if they are equally distant from the center.

G.116 diameter is the greatest chord

Of chords in a circle, the diameter is the greatest, and a chord nearer the center is always greater than one more remote.

G.118 perpendicular at extremity of a diameter is tangent

The line drawn at right angles to the diameter of a circle from its extremity falls outside the circle, no other line can be interposed between it and the arc, and the angle of the semicircle is greater than any acute rectilineal angle.

G.119 draw a tangent to a circle from a point

From a given point, construct a line tangent to a given circle.

G.120 radius to point of contact is perpendicular to tangent

If a line touches a circle, the segment joined from the center to the point of contact is perpendicular to the tangent.

G.121 perpendicular to tangent at contact passes through center

If a line touches a circle, the line drawn at right angles to the tangent from the point of contact passes through the center of the circle.

G.123 inscribed angle theorem

In a circle, the angle at the center is double the angle at the circumference when the angles have the same arc as base.

G.124 angles in the same circle segment are equal

In a circle, angles in the same segment are equal to one another.

G.129 similar circle segments on equal chords are equal

Similar segments of circles on equal segments are equal to one another.

G.130 complete a circle from a given segment

Given a segment of a circle, complete the full circle of which it is a segment.

G.133 equal angles stand on equal arcs

In equal circles, equal angles stand on equal arcs, whether they stand at the centers or at the circumferences.

G.134 angles standing on equal arcs are equal

In equal circles, angles standing on equal arcs are equal to one another, whether they stand at the centers or at the circumferences.

G.135 equal chords cut off equal arcs

In equal circles, equal chords cut off equal arcs, the greater equal to the greater and the less to the less.

G.136 equal arcs are subtended by equal chords

In equal circles, equal arcs are subtended by equal chords.

G.137 bisect an arc

Bisect a given arc of a circle.

G.138 angle in a circle segment

An:term:**angle in a segment** <angle-in-segment> is the:term:angle <G.22> which, when a:term:point <G.1> is taken on the arc of the segment and:term:segments <G.4> are joined from it to the:term:extremities <G.6> of the segment which is the:term:**base of the segment** <base-of-segment>, is contained by the segments so joined.

G.139 Thales’ theorem and angles in circle segments

In a circle, the angle in a semicircle is right, the angle in a greater segment is acute, and the angle in a lesser segment is obtuse.

G.140 alternate segment theorem

If a line touches a circle and a chord is drawn from the point of contact, the angles which the chord makes with the tangent equal the angles in the alternate segments of the circle.

G.141 construct a circle segment on a chord admitting a given angle

On a given segment, construct a segment of a circle admitting an angle equal to a given rectilineal angle.

G.142 cut off a circle segment admitting a given angle

From a given circle, cut off a segment admitting an angle equal to a given rectilineal angle.

G.143 intersecting chords theorem

If two chords in a circle intersect one another, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.

G.144 tangent-secant theorem

If from a point outside a circle a secant and a tangent are drawn, the rectangle contained by the whole secant and its external segment equals the square on the tangent.

G.145 converse of the tangent-secant theorem

If from a point outside a circle a secant and a segment to the circle are drawn such that the rectangle on the whole secant and its external part equals the square on the segment, the segment is tangent to the circle.

G.147 fit a chord in a circle equal to a segment

Fit into a given circle a chord equal to a given segment not greater than the diameter of the circle.

G.149 inscribe a triangle in a circle equiangular with a triangle

Inscribe in a given circle a triangle equiangular with a given triangle.

G.151 circumscribe a triangle about a circle equiangular with a triangle

Circumscribe about a given circle a triangle equiangular with a given triangle.

G.154 inscribe a circle in a triangle

Inscribe a circle in a given triangle.

G.157 circumscribe a circle about a triangle

Circumscribe a circle about a given triangle.

G.158 inscribe a square in a circle

Inscribe a square in a given circle.

G.159 circumscribe a square about a circle

Circumscribe a square about a given circle.

G.160 inscribe a circle in a square

Inscribe a circle in a given square.

G.161 circumscribe a circle about a square

Circumscribe a circle about a given square.

G.162 construct a golden isosceles triangle

Construct an isosceles triangle having each of the angles at the base double the remaining angle.

G.164 circumscribe a regular pentagon about a circle

Circumscribe an equilateral and equiangular pentagon about a given circle.

G.165 inscribe a circle in a regular pentagon

Inscribe a circle in a given equilateral and equiangular pentagon.

G.166 circumscribe a circle about a regular pentagon

Circumscribe a circle about a given equilateral and equiangular pentagon.

G.167 inscribe a regular hexagon in a circle

Inscribe an equilateral and equiangular hexagon in a given circle.

G.168 inscribe a regular pentadecagon in a circle

Inscribe an equilateral and equiangular fifteen-angled figure in a given circle.

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.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.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.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.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.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.195 magnitudes proportional proportional

If four magnitudes be proportional, they is proportional alternately.

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.208 magnitudes proportional greatest

If four magnitudes be proportional, the greatest and the least are greater than the remaining two.

G.210 triangles parallelograms height

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

G.211 segment drawn parallel

If a segment be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section is parallel to the remaining side of the triangle.

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.213 equiangular triangles sides

In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.

G.214 triangles sides proportional

If two triangles have their sides proportional, the triangles is equiangular and will have those angles equal which the corresponding sides subtend.

G.215 triangles angle equal

If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles is equiangular and will have those angles equal which the corresponding sides subtend.

G.216 triangles angle equal

If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles is equiangular and will have those angles equal, the sides about which are proportional.

G.218 right angled triangle

If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.

G.219 cut nth part

From a given segment to cut off a prescribed part.

G.220 cut similarly

Cut a given uncut segment similarly to a given cut segment.

G.221 third proportional

To two given segments to find a third proportional.

G.222 fourth proportional

To three given segments to find a fourth proportional.

G.223 mean proportional

To two given segments to find a mean proportional.

G.227 segments proportional rectangle

If four segments be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, the four segments is proportional.

G.228 segments proportional rectangle

If three segments be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, the three segments is proportional.

G.229 describe on a given segment a rectilinear figure

Describe on a given segment a rectilinear figure similar and similarly situated to a given rectilinear figure.

G.233 figures similar polygon

Figures which are similar to the same polygon are also similar to one another.

G.234 segments proportional polygons

If four segments be proportional, the polygons similar and similarly described upon them is proportional; and, if the polygons similar and similarly described upon them be proportional, the segments will themselves also be proportional.

G.235 equiangular parallelograms ratio

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

G.238 parallelogram there taken

If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole.

G.239 parallelograms applied segment

Of all the parallelograms applied to the same segment and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the segment, that parallelogram is greatest which is applied to the half of the segment and is similar to the defect.

G.240 segment apply parallelogram

To a given segment to apply a parallelogram equal to a given polygon and deficient by a parallelogrammic figure similar to a given one: thus the given polygon must not be greater than the parallelogram described on the half of the segment and similar to the defect.

G.241 segment apply parallelogram

To a given segment to apply a parallelogram equal to a given polygon and exceeding by a parallelogrammic figure similar to a given one.

G.243 extreme and mean ratio

Cut a given segment in extreme and mean ratio.

G.244 right angled triangles

In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

G.245 triangles sides proportional

If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles is in a segment.

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.250 unequal numbers set

Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers is prime to one another.

G.251 numbers prime find

Given two numbers not prime to one another, to find their greatest common measure.

G.252 numbers prime find

Given three numbers not prime to one another, to find their greatest common measure.

G.255 number part parts

Any number is either a part or parts of any number, the less of the greater.

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.258 number parts number

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

G.261 number part number

If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

G.262 number parts number

If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.

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.266 numbers proportional proportional

If four numbers be proportional, they is proportional alternately.

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.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.270 numbers multiplying make

If two numbers by multiplying one another make certain numbers, the numbers so produced are equal.

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.273 numbers proportional number

If four numbers be proportional, the number produced from the first and fourth will equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, the four numbers is proportional.

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.277 numbers prime number

If two numbers be prime to one another, the number which measures the one of them is prime to the remaining number.

G.278 numbers prime number

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

G.279 numbers prime product

If two numbers be prime to one another, the product of one of them into itself is prime to the remaining one.

G.280 numbers prime numbers

If two numbers be prime to two numbers, both to each, their products also is prime to one another.

G.281 numbers prime multiplying

If two numbers be prime to one another, and each by multiplying itself make a certain number, the products is prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter is prime to one another [and this is always the case with the extremes].

G.282 numbers prime sum

If two numbers be prime to one another, the sum is prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers is prime to one another.

G.284 prime number prime

Any prime number is prime to any number which it does not measure.

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.287 composite number measured

Any composite number is measured by some prime number.

G.288 number prime measured

Any number either is prime or is measured by some prime number.

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.290 numbers find least

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

G.291 numbers measure number

If two numbers measure any number, the least number measured by them will also measure the same.

G.292 numbers find least

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

G.293 number measured number

If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.

G.294 number part whatever

If a number have any part whatever, it is measured by a number called by the same name as the part.

G.295 find the number which is the least that

Find the number which is the least that will have given parts.

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.303 there many numbers

If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.

G.304 there many numbers

If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.

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.307 numbers fall numbers

If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves 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.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.316 square number measure

If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.

G.317 cube number measure

If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube 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.322 mean proportional number

If one mean proportional number fall between two numbers, the numbers is similar plane numbers.

G.323 mean proportional numbers

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

G.324 numbers continued proportion

If three numbers be in continued proportion, and the first be square, the third is square.

G.325 numbers continued proportion

If four numbers be in continued proportion, and the first be cube, the fourth is cube.

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.330 similar plane numbers

If two similar plane numbers by multiplying one another make some number, the product is square.

G.331 numbers multiplying make

If two numbers by multiplying one another make a square number, they are similar plane numbers.

G.332 cube number multiplying

If a cube number by multiplying itself make some number, the product is cube.

G.333 cube number multiplying

If a cube number by multiplying a cube number make some number, the product is cube.

G.334 cube number multiplying

If a cube number by multiplying any number make a cube number, the multiplied number is cube.

G.335 number multiplying itself

If a number by multiplying itself make a cube number, it will itself also be cube.

G.336 composite number multiplying

If a composite number by multiplying any number make some number, the product is solid.

G.337 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit is square, as will also those which successively leave out one; the fourth is cube, as will also all those which leave out two; and the seventh is at once cube and square, as will also those which leave out five.

G.338 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 square, all the rest is square. And, if the number after the unit be cube, all the rest is cube.

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.340 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the proportional 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.343 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 prime, the greatest will not be measured by any except those which have a place among the proportional numbers.

G.344 number least measured

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

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.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.348 numbers investigate whether

Given two numbers, to investigate whether it is possible to find a third proportional to them.

G.349 numbers investigate possible

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

G.350 prime numbers more

Prime numbers are more than any assigned multitude of prime numbers.

G.359 odd number multiplying

If an odd number by multiplying an even number make some number, the product is even.

G.360 odd number multiplying

If an odd number by multiplying an odd number make some number, the product is odd.

G.361 odd number measure

If an odd number measure an even number, it will also measure the half of it.

G.362 odd number prime

If an odd number be prime to any number, it is prime to the double of it.

G.364 numbers continually doubled

Each of the numbers which are continually doubled beginning from a dyad is even-times even only.

G.366 number half odd

If a number have its half odd, it is even-times odd only.

G.367 number continually doubled

If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both even-times even and even-times odd.

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.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.373 less unequal magnitudes

If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes is incommensurable.

G.374 commensurable magnitudes find

Given two commensurable magnitudes, to find their greatest common measure.

G.375 commensurable magnitudes find

Given three commensurable magnitudes, to find their greatest common measure.

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.385 magnitudes commensurable incommensurable

If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one is incommensurable with the same.

G.386 segments proportional square

If four segments be proportional, and the square on the first be greater than the square on the second by the square on a segment commensurable with the first, the square on the third is greater than the square on the fourth by the square on a segment commensurable with the third. And, if the square on the first be greater than the square on the second by the square on a segment incommensurable with the first, the square on the third is greater than the square on the fourth by the square on a segment incommensurable with the third.

G.387 commensurable magnitudes added

If two commensurable magnitudes be added together, the whole is commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes is commensurable.

G.388 incommensurable magnitudes added

If two incommensurable magnitudes be added together, the whole is incommensurable with each of them; and, if the whole be incommensurable with one of them, the original magnitudes is incommensurable.

G.389 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, then the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment commensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are commensurable in length.

G.390 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater is greater than the square on the less by the square on a segment incommensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.

G.396 square medial segment

The square on a medial segment, if applied to a rational segment, produces as breadth a segment rational and incommensurable in length with that to which it is applied.

G.397 segment commensurable medial

A segment commensurable with a medial segment is medial.

G.401 find medial segments commensurable in square only which

Find medial segments commensurable in square only which contain a rational rectangle.

G.402 find medial segments commensurable in square only which

Find medial segments commensurable in square only which contain a medial rectangle.

G.405 find two medial segments commensurable in square only

Find two medial segments commensurable in square only, containing a rational rectangle, 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.406 find two medial segments commensurable in square only

Find two medial segments commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater.

G.407 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.

G.408 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

G.416 binomial segment divided

A binomial segment is divided into its terms at one point only.

G.417 bimedial segment divided

A first bimedial segment is divided at one point only.

G.418 bimedial segment divided

A second bimedial segment is divided at one point only.

G.419 major segment divided

A major segment is divided at one and the same point only.

G.420 side rational plus

The side of a rational plus a medial area is divided at one point only.

G.421 side sum medial

The side of the sum of two medial areas is divided at one point only.

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.434 area contained rational

If an area be contained by a rational segment and the first binomial, the “side” of the area is the irrational segment which is called binomial.

G.435 area contained rational

If an area be contained by a rational segment and the second binomial, the “side” of the area is the irrational segment which is called a first bimedial.

G.436 area contained rational

If an area be contained by a rational segment and the third binomial, the “side” of the area is the irrational segment called a second bimedial.

G.437 area contained rational

If an area be contained by a rational segment and the fourth binomial, the “side” of the area is the irrational segment called major.

G.438 area contained rational

If an area be contained by a rational segment and the fifth binomial, the “side” of the area is the irrational segment called the side of a rational plus a medial area.

G.439 area contained rational

If an area be contained by a rational segment and the sixth binomial, the “side” of the area is the irrational segment called the side of the sum of two medial areas.

G.440 square binomial segment

The square on the binomial segment applied to a rational segment produces as breadth the first binomial.

G.441 square bimedial segment

The square on the first bimedial segment applied to a rational segment produces as breadth the second binomial.

G.442 square bimedial segment

The square on the second bimedial segment applied to a rational segment produces as breadth the third binomial.

G.443 square major segment

The square on the major segment applied to a rational segment produces as breadth the fourth binomial.

G.444 square side rational

The square on the side of a rational plus a medial area applied to a rational segment produces as breadth the fifth binomial.

G.445 square side sum

The square on the side of the sum of two medial areas applied to a rational segment produces as breadth the sixth binomial.

G.446 segment commensurable length

A segment commensurable in length with a binomial segment is itself also binomial and the same in order.

G.447 segment commensurable length

A segment commensurable in length with a bimedial segment is itself also bimedial and the same in order.

G.448 segment commensurable major

A segment commensurable with a major segment is itself also major.

G.449 segment commensurable side

A segment commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

G.450 segment commensurable side

A segment commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.

G.451 rational medial area

If a rational and a medial area be added together, four irrational segments arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.

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.477 area contained rational

If an area be contained by a rational segment and a first apotome, the “side” of the area is an apotome.

G.478 area contained rational

If an area be contained by a rational segment and a second apotome, the “side” of the area is a first apotome of a medial segment.

G.479 area contained rational

If an area be contained by a rational segment and a third apotome, the “side” of the area is a second apotome of a medial segment.

G.480 area contained rational

If an area be contained by a rational segment and a fourth apotome, the “side” of the area is minor.

G.481 area contained rational

If an area be contained by a rational segment and a fifth apotome, the “side” of the area is a segment which produces with a rational area a medial whole.

G.482 area contained rational

If an area be contained by a rational segment and a sixth apotome, the “side” of the area is a segment which produces with a medial area a medial whole.

G.483 square apotome applied

The square on an apotome applied to a rational segment produces as breadth a first apotome.

G.484 square apotome medial

The square on a first apotome of a medial segment applied to a rational segment produces as breadth a second apotome.

G.485 square apotome medial

The square on a second apotome of a medial segment applied to a rational segment produces as breadth a third apotome.

G.486 square minor segment

The square on a minor segment applied to a rational segment produces as breadth a fourth apotome.

G.487 square segment produces

The square on the segment which produces with a rational area a medial whole, if applied to a rational segment, produces as breadth a fifth apotome.

G.488 square segment produces

The square on the segment which produces with a medial area a medial whole, if applied to a rational segment, produces as breadth a sixth apotome.

G.497 apotome binomial segment

The apotome is not the same with the binomial segment.

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.501 medial segment there

From a medial segment there arise irrational segments infinite in number, and none of them is the same as any of the preceding.

G.502 part segment plane

A part of a segment cannot be in the plane of reference and a part in a plane more elevated.

G.503 segments cut plane

If two segments cut one another, they are in one plane, and every triangle is in one plane.

G.504 planes cut common

If two planes cut one another, their common section is a segment.

G.506 segment set right

If a segment be set up at right angles to two segments which cut one another, at their common point of section, it is at right angles to the plane through them.

G.507 segment set right

If a segment be set up at right angles to three segments which meet one another, at their common point of section, the three segments are in one plane.

G.508 segments right angles

If two segments be at right angles to the same plane, the segments is parallel.

G.509 segments parallel points

If two segments be parallel and points be taken at random on each of them, the segment joining the points is in the same plane with the parallel lines.

G.510 segments parallel right

If two segments be parallel, and one of them be at right angles to any plane, the remaining one is at right angles to the same plane.

G.511 straight lines parallel

Straight lines which are parallel to the same segment and are not in the same plane with it are also parallel to one another.

G.513 elevated point draw

From a given elevated point to draw a segment perpendicular to a given plane.

G.514 set segment right

To set up a segment at right angles to a given plane from a given point in it.

G.515 point segments set

From the same point two segments cannot be set up at right angles to the same plane on the same side.

G.517 planes segment right

Planes to which the same segment is at right angles is parallel.

G.518 segments meeting parallel

If two segments meeting one another be parallel to two segments meeting one another, not being in the same plane, the planes through them are parallel.

G.519 parallel planes cut

If two parallel planes be cut by any plane, their common sections are parallel.

G.520 segments cut parallel

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

G.522 segment right angles

If a segment be at right angles to any plane, all the planes through it is at right angles to the same plane.

G.523 planes cut right

If two planes which cut one another be at right angles to any plane, their common section is at right angles to the same plane.

G.524 solid angle

A:term:**solid angle** <solid-angle> is the inclination constituted by more than two:term:lines <G.2> which meet one another and are not in the same:term:surface <G.533>, towards all the lines. Otherwise: A:term:**solid angle** <solid-angle> is that which is contained by more than two:term:plane angles <G.22> which are not in the same:term:plane <G.7> and are constructed to one:term:point <G.1>.

G.525 solid angle contained

If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.

G.526 solid angle contained

Any solid angle is contained by plane angles less than four right angles.

G.527 there plane angles

If there be three plane angles of which two, taken together in any manner, are greater than the remaining one, and they are contained by equal segments, it is possible to construct a triangle out of the segments joining the extremities of the equal segments.

G.528 construct a solid angle out of three plane

Construct a solid angle out of three plane angles two of which, taken together in any manner, are greater than the remaining one: thus the three angles must be less than four right angles.

G.532 segment point construct

On a given segment, and at a given point on it, to construct a solid angle equal to a given solid angle.

G.536 describe on a given segment a parallelepipedal solid

Describe on a given segment a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid.

G.540 parallelepipedal solids base

Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are not on the same segments, are equal.

G.541 parallelepipedal solids equal

Parallelepipedal solids which are on equal bases and of the same height are equal.

G.544 equal parallelepipedal solids

In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal.

G.545 there equal plane

If there be two equal plane angles, and on their vertices there be set up elevated segments containing equal angles with the original segments respectively, if on the elevated segments points be taken at random and perpendiculars be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes segments be joined to the vertices of the original angles, they will contain, with the elevated segments, equal angles.

G.546 segments proportional parallelepipedal

If three segments be proportional, the parallelepipedal solid formed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.

G.549 sides opposite planes

If the sides of the opposite planes of a cube be bisected, and planes be carried through the points of section, the common section of the planes and the diameter of the cube bisect one another.

G.552 circles squares diameters

Circles are to one another as the squares on the diameters.

G.553 pyramid

A:term:**pyramid** <pyramid> is a:term:solid figure <G.529>, contained by:term:planes <G.7>, which is constructed from one plane to one:term:point <G.1>.

G.554 pyramid triangular base

Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.

G.555 there pyramids height

If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two equal prisms, then, as the base of the one pyramid is to the base of the other pyramid, so will all the prisms in the one pyramid be to all the prisms, being equal in multitude, in the other pyramid.

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