The Elements of Euclid: Viz. the First Six Books, with the Eleventh and Twelfth. In which the Corrections of Dr. Simson are Generally Adopted, But the Errors Overlooked by Him are Corrected, and the Obscurities of His and Other Editions Explained. Also Some of Euclid's Demonstrations are Restored, Others Made Shorter and More General, and Several Useful Propositions are Added. Together with Elements of Plane and Spherical Trigonometry, and a Treatise on Practical Geometry |
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Page 16
... bafe BC fhail be equal to the bafe EF ; and the triangle ABC to the tri- angle DEF ; and the other angles , to which the equal fides are oppofite , shall be equal each to each , viz . the angle ABC to the angle DEF , and the angle ACB ...
... bafe BC fhail be equal to the bafe EF ; and the triangle ABC to the tri- angle DEF ; and the other angles , to which the equal fides are oppofite , shall be equal each to each , viz . the angle ABC to the angle DEF , and the angle ACB ...
Page 17
... bafe FC is equal b to the base GB , and the triangle AFC to the triangle AGB ; and the remaining angles of the one are equal b to the remaining angles of the other , each to each , to which the equal fides are oppofite , viz . the angle ...
... bafe FC is equal b to the base GB , and the triangle AFC to the triangle AGB ; and the remaining angles of the one are equal b to the remaining angles of the other , each to each , to which the equal fides are oppofite , viz . the angle ...
Page 18
... bafe AB , and upon the fame fide of it , which have their fides CA , DA , terminated in the extremity A of the base , equal to one another ; their fides CB , DB , that are terminated in B , fhall not be equal . Join CD ; then in the ...
... bafe AB , and upon the fame fide of it , which have their fides CA , DA , terminated in the extremity A of the base , equal to one another ; their fides CB , DB , that are terminated in B , fhall not be equal . Join CD ; then in the ...
Page 19
... bafe BC equal to the bafe EF . The angle BAC is equal to the angle EDF . For , if the triangle ABC be applied to DEF , fo that the point B be on E , and the ftraight line BC upon EF ; the point C fhall alfo coincide with the point F , B ...
... bafe BC equal to the bafe EF . The angle BAC is equal to the angle EDF . For , if the triangle ABC be applied to DEF , fo that the point B be on E , and the ftraight line BC upon EF ; the point C fhall alfo coincide with the point F , B ...
Page 20
... bafe DF is alfo B C cs . 1. equal to the bafe EF ; therefore the angle DAF is equal © to the angle EAF ; wherefore the given rectilineal angle BAC is bifected by the ftraight line AF . Which was to be done . a I. I. b 9. I. € 4. I. 23.1 ...
... bafe DF is alfo B C cs . 1. equal to the bafe EF ; therefore the angle DAF is equal © to the angle EAF ; wherefore the given rectilineal angle BAC is bifected by the ftraight line AF . Which was to be done . a I. I. b 9. I. € 4. I. 23.1 ...
Common terms and phrases
ABC is equal ABCD alfo alſo angle ABC angle ACB angle BAC arch bafe baſe becauſe the angle bifect Book XI cafe centre circle ABC circumference cofine confequently cylinder defcribed demonftrated diameter equal angles equiangular equimultiples Euclid exterior angle faid fame altitude fame manner fame multiple fame number fame ratio fame reaſon fecond fegment fhall fides fimilar firft firſt folid angle fome fore fquare fquare of AC fuperficies given ftraight line gnomon greater half the fum join lefs leſs Let ABC magnitudes meaſure oppofite angle pafs parallel parallelogram parallelopiped perpendicular plane angles prifm PROB propofition proportionals Q. E. D. PROP radius rectangle contained rectilineal figure remaining angle right angles ſhall ſquare tangent thefe THEOR theſe tiple triangle ABC Wherefore
Popular passages
Page 30 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
Page 142 - 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 13 - Let it be granted that a straight line may be drawn from any one point to any other point.
Page 30 - ... then shall the other sides 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, BCA equal to the angles DEF, EFD, viz.
Page 72 - The diameter is the greatest straight line in a circle; and of all others, that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less. Let ABCD be a circle, of which...
Page 57 - If then the sides of it, BE, ED are equal to one another, it is a square, and what was required is now done: But if they are not equal, produce one of them BE to F, and make EF equal to ED, and bisect BF in G : and from the centre G, at the distance GB, or GF, describe the semicircle...
Page 145 - AC the same multiple of AD, that AB is of the part which is to be cut off from it : join BC, and draw DE parallel to it : then AE is the part required to be cut off.
Page 48 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.
Page 35 - F, which is the common vertex of the triangles ; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 10 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.