## Euclid's Elements of Geometry: The Six First Books. To which are Added, Elements of Plain and Spherical Trigonometry, a System of Conick Sections, Elements of Natural Philosophy, as Far as it Relates to Astronomy, According to the Newtonian System, and Elements of Astronomy: with Notes |

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Page 23

but the angle ACD is greater than its part BCD A [ Ax . 9 ] , therefore the angle

ADC is greater than BCD , of

[ Ax . 9 ) , is greater than BCD : but , because the sides BC , BD of the triangle

BDC ...

but the angle ACD is greater than its part BCD A [ Ax . 9 ] , therefore the angle

ADC is greater than BCD , of

**course**, the angle BDČ , which is greater than ADC[ Ax . 9 ) , is greater than BCD : but , because the sides BC , BD of the triangle

BDC ...

Page 24

... and the bases DF , EF also equal [ Constr . ] ; therefore the angles DAF , EAF

are equal [ 8. 1 ] , and of

AF . D Cor . - By the aid of this proposition , 24 BOOK I. EUCLID'S ELEMENTS.

... and the bases DF , EF also equal [ Constr . ] ; therefore the angles DAF , EAF

are equal [ 8. 1 ] , and of

**course**the given angle BAC is bisected by the right lineAF . D Cor . - By the aid of this proposition , 24 BOOK I. EUCLID'S ELEMENTS.

Page 26

A like absurdity would follow , if AB were to fall on the other side of DE towards F.

Therefore AB coincides with DE , and so the angles BAC , EDF coincide , and are

of

A like absurdity would follow , if AB were to fall on the other side of DE towards F.

Therefore AB coincides with DE , and so the angles BAC , EDF coincide , and are

of

**course**equal [ Ax . 8 ] . PROP . XII . PROB . From a given point ( C ) , without ... Page 27

1 ) , and of

The angles ( ABC , ABD ) , which one right line ( AB ) , makes with another ( CD )

, on one side of it , are together equal to two right angles . Fig . 2 Fig.2 . B B If the

...

1 ) , and of

**course**CG is perpendicular to AB ( Def . 20 ) . PROP . XIII . THEOR .The angles ( ABC , ABD ) , which one right line ( AB ) , makes with another ( CD )

, on one side of it , are together equal to two right angles . Fig . 2 Fig.2 . B B If the

...

Page 30

Because the triangle ABD is isosceles , the angles ABD , ADB are equal ( 5. 1 ] ;

but the external angle ADB of the triangle BDC is greater than the internal remote

angle C , ( 16. 1 ) , therefore ABD is greater than C ; of

Because the triangle ABD is isosceles , the angles ABD , ADB are equal ( 5. 1 ] ;

but the external angle ADB of the triangle BDC is greater than the internal remote

angle C , ( 16. 1 ) , therefore ABD is greater than C ; of

**course**ABC , which is ...### What people are saying - Write a review

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Euclid's Elements of Geometry, the First Six Books: To Which Are Added ... John Allen No preview available - 2015 |

Euclid's Elements of Geometry, the First Six Books: To Which Are Added ... John Allen No preview available - 2017 |

### Common terms and phrases

adding applied arch axis base bisected body centre circle circumference common compounded conick section Constr contained contrary course described diameter difference directrix distance double draw drawn ellipse equal equal angles equiangular extremes figure focus force formed four given greater half hyperbola inscribed join legs less let fall magnitudes manner meet motion opposite ordinate parabola parallel parallelogram parameter passing perpendicular plain principal produced PROP proportional proposition proved radius ratio rectangle remaining right angles right line secant segments shewn sides similar sine square taken tangent THEOR third touching triangle ABC unequal vertex whence whole

### Popular passages

Page 40 - Therefore all the interior angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 428 - Lastly, if it universally appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that in proportion to the quantity of matter which they severally contain: that the moon likewise, according to the quantity of its matter, gravitates towards the earth; that, on the other hand, our sea gravitates towards the moon; and all the planets mutually one towards another; and the comets in like manner towards the sun...

Page 13 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.

Page 116 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.

Page 430 - A stone, whirled about in a sling, endeavors to recede from the hand that turns it; and by that endeavor, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as it is let go, flies away.

Page 374 - ... figure, together with four right angles, are equal to twice as many right angles as the figure has be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of its angles.

Page 459 - In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance...

Page 434 - Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say...

Page 127 - D, is said to be Compounded of the ratios of the first to the second, of the second to the third, and so on to the last.

Page 106 - A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle. 5. In like manner, a circle is said to be inscribed...