## The Elements of Euclid: Also the Book of Euclid's Data ...cor. viz. The first six books, together with the eleventh and twelfthF. Wingrave, 1804 - Trigonometry |

### From inside the book

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... , are equal to one another . VIII . Magnitudes , which coincide with one another , that is , which exactly fill the same

... , are equal to one another . VIII . Magnitudes , which coincide with one another , that is , which exactly fill the same

**space**, are equal to one another . Book I. Took I. IX . The whole is greater than its A 3 OF EUCLID . 5. Page 6

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**space**. ΧΙ . All right angles are equal to one another . XII . " If a straight line meets two straight lines , so as to make the " two interior angles on the same side of it taken together less " than two right angles , these straight ... Page 9

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**space**, which is impossible . Therefore the base BC shall coincide with the a . 1o . Ax . base EF , and be equal to it . Wherefore the whole triangle ABC shall coincide with the whole triangle DEF , and be equal to it ; and the other ... Page 192

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**space**betwixt them ; which a . 10. Ax.1 . is impoffible a . therefore BD the common section of the planes AB , BC cannot but be a straight line . Wherefore if two planes , & c . Q. E. D. See N. a . 15. 1 . b . 4. I. с . 26. 1 . F Γ PROP ... Page 195

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**space**be- twixt them , which is impossible a . a . 10 Ax.1 . Therefore the straight line joining C FD the points E , F is not above the plane in which the parallels AB , CD are , and is therefore in that plane . Wherefore if two ...### Common terms and phrases

alſo altitude angle ABC angle BAC baſe BC BC is equal BC is given becauſe the angle becauſe the ratio biſected Book XI cafe cauſe circle ABCD circumference cone conſtruction cylinder demonſtrated deſcribed diameter drawn EFGH equal angles equiangular equimultiples Euclid exceſs fame multiple fame ratio fame reaſon fides fimilar firſt folid angle fore given angle given in magnitude given in poſition given in ſpecies given magnitude given ratio given ſtraight line gnomon greater join leſs likewife oppoſite parallel parallelepipeds parallelogram paſs perpendicular plane angles priſm PROP proportionals Propoſition pyramid rectangle contained rectilineal figure right angles ſame ſecond ſegment ſhall ſhall be equal ſhewn ſide ſolid ſome ſpace ſphere ſquare of AC ſtraight line AC THEOR theſe thoſe thro triangle ABC vertex wherefore

### Popular passages

Page 157 - D ; wherefore the remaining angle at C is equal to the remaining angle at F ; Therefore the triangle ABC is equiangular to the triangle DEF. Next, let each of the angles at C, F be not less than a right angle ; the triangle ABC is also, in this case, equiangular to the triangle DEF.

Page 24 - ... be equal, each to each ; and also the third angle of the one to the third angle of the other. Let ABC, DEF be two triangles which have the angles ABC, ECA equal to the angles DEF, EFD, viz.

Page 1 - The circumference of every circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, and these into thirds, fourths, &c.

Page 45 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Page 73 - THE straight line drawn at right angles to the diameter of a circle, from the extremity of...

Page 163 - 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 81 - ELF. Join BC, EF ; and because the circles ABC, DEF are equal, the straight lines drawn from their centres are equal: therefore the two sides BG, GC, are equal to the two EH, HF ; and the angle at G is equal to the angle at H ; therefore the base BC is equal (4.

Page 323 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Page 80 - EA : and because AD is equal to DC, and DE common to the triangles ADE, CDE, the two sides AD, DE are equal to the two CD, DE, each to each ; and the angle ADE is equal to the angle CDE, for each of them is a right angle ; therefore the base AE is equal (4.

Page 36 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.