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 24
... ACB ; but ACD , ACB are together equal b to two right angles ; therefore the angles ABC , BCA are less than two right angles . In like manner , it may be demonstrated , that BAC , : BAC , ACB , as alfo CAB , ABC are 24 THE ELEMENTS.
... ACB ; but ACD , ACB are together equal b to two right angles ; therefore the angles ABC , BCA are less than two right angles . In like manner , it may be demonstrated , that BAC , : BAC , ACB , as alfo CAB , ABC are 24 THE ELEMENTS.
Page 26
... demonstrated , that the fides AB , BC are greater than CA , and BC , CA greater than AB . Therefore any two fides , & c . Q : E. D. C 19. I. IF PROP . XXI , THEOR . [ F , from the ends of the fide of a triangle , there be drawn two ...
... demonstrated , that the fides AB , BC are greater than CA , and BC , CA greater than AB . Therefore any two fides , & c . Q : E. D. C 19. I. IF PROP . XXI , THEOR . [ F , from the ends of the fide of a triangle , there be drawn two ...
Page 39
... demonstrated , that no other line but AD is parallel to BC ; AD is therefore parallel to it . Wherefore equal triangles upon , & c . Q. E. D. A B E D C COR . Hence it is manifeft , that the ftraight line , which meets another ftraight ...
... demonstrated , that no other line but AD is parallel to BC ; AD is therefore parallel to it . Wherefore equal triangles upon , & c . Q. E. D. A B E D C COR . Hence it is manifeft , that the ftraight line , which meets another ftraight ...
Page 44
... demonstrated , that it is equilateral ; it is therefore a square , and it is defcribed upon the given straight line AB . Which was to be done . D A COR . Hence every parallelogram that has one right angle , has all its angles right ...
... demonstrated , that it is equilateral ; it is therefore a square , and it is defcribed upon the given straight line AB . Which was to be done . D A COR . Hence every parallelogram that has one right angle , has all its angles right ...
Page 101
... demonstrated , that the angles ABC , BCD , B CDA are severally bifected by the straight lines BD , AC ; therefore , because the b BOOK IV . a 8. I. angle DAB is equal to the angle ABC , and that the angle EAB is the half of DAB , and ...
... demonstrated , that the angles ABC , BCD , B CDA are severally bifected by the straight lines BD , AC ; therefore , because the b BOOK IV . a 8. I. angle DAB is equal to the angle ABC , and that the angle EAB is the half of DAB , and ...
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.