## The Elements of Euclid: viz. the first six books, together with the eleventh and twelfth; and also the book of Euclid's Data |

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Page 241

If there be two triangular

parallelogram, and the base of the other a triangle; if the parallelogram be double

the triangle, the

If there be two triangular

**prisms**of the same allitude, the base of one of which is aparallelogram, and the base of the other a triangle; if the parallelogram be double

the triangle, the

**prisms**shall be equal to one another. Let the**prisms**ABCDEF, ... Page 249

another, therefore the

and the straight lines KH opposite to it, is equal S-7 to the

triangle GFC for its base, and the triangle HKL opposite to it; for they are of the

same ...

another, therefore the

**prism**having the parallelogram EBFG Book XII. for its base,and the straight lines KH opposite to it, is equal S-7 to the

**prism**having thetriangle GFC for its base, and the triangle HKL opposite to it; for they are of the

same ...

Page 250

Let there be two pyramids of the same altitude upon the triangular bases ABC,

DEF, and having their vertices in the points G, H ; and let each of them be divided

into two equal pyramids, similar to the whole, and into two equal

Let there be two pyramids of the same altitude upon the triangular bases ABC,

DEF, and having their vertices in the points G, H ; and let each of them be divided

into two equal pyramids, similar to the whole, and into two equal

**prisms**; and let ... Page 251

RVFSTY; so is the

ABC to the B base DEF: Therefore, as the base ABC to the base DEF, so are the

two ...

RVFSTY; so is the

**prism**LXCOMN to the**prism**Book xii. RVFSTY: But as the**prism**LXCOMN to the**prism**S-- RVFSTY, so is, as has been proved, the baseABC to the B base DEF: Therefore, as the base ABC to the base DEF, so are the

two ...

Page 259

Upon the square ABCD erect a

the circle, and a

Upon the square ABCD erect a

**prism**of the same altitude with the cylinder; this**prism**is greater than half the cylinder; because if a square be described aboutthe circle, and a

**prism**erected upon the square, of the same altitude with the ...### What people are saying - Write a review

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### Common terms and phrases

ABC is given ABCD altitude angle ABC angle BAC arch base BC BC is equal bisected Book XI centre circle ABC circumference cone cosine cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gles gnomon greater half the perimeter hypotenuse join less Let ABC multiple parallel parallelogram perpendicular point F polygon prism proportionals proposition Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment side BC similar sine solid angle solid parallelepiped square of AC straight line AB straight line BC tangent THEOR third tiple triangle ABC vertex wherefore

### Popular passages

Page 43 - If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced. Let the straight line AB be bisected in C, and produced to D : the rectangle AD, DB, together with the square of CB, shall be equal to the square of CD.

Page 304 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Page 26 - if a straight line," &c. QED PROP. XXIX. THEOR. See the Jf a straight line fall upon two parallel straight ti?isepropo- lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two right angles.

Page 52 - In every triangle, the square on the side subtending either of the acute angles, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the...

Page 168 - EQUIANGULAR parallelograms have to one another the ratio which is compounded of the ratios of their sides.* Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG : the ratio of the parallelogram AC to the parallelogram CF, is the same with the ratio which is compounded of the ratios of their sides. • See Note. Let BG, CG, be placed in a straight line ; therefore DC and CE are also in a straight line (14.

Page 151 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 30 - And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB...

Page 28 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 62 - ... than the more remote: but of those which fall upon the convex circumference, the least is that between the point without the circle and the diameter; and, of the rest, that which is nearer to the least is always less than the more remote: and only two equal straight lines can be drawn from the point into the circumference, one upon each side of the least.

Page 5 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...