## The First Six Books of the Elements of Euclid, with a Commentary and Geometrical Exercises: To which are Annexed a Treatise on Solid Geometry, and a Short Essay on the Ancient Geometrical AnalysisJohn Taylor, 30 Upper Gower Street, Bookseller and Publisher to the University: and sold, 1828 - Euclid's Elements - 324 pages |

### From inside the book

Page 79

... and subtract half the difference from half the sum , and the remainder is the

less . ( 262 ) Since

follows that if the area be divided by one side the quote will be the other side .

... and subtract half the difference from half the sum , and the remainder is the

less . ( 262 ) Since

**the area of a rectangle is equal to the product of its**sides , itfollows that if the area be divided by one side the quote will be the other side .

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### Common terms and phrases

A B C altitude appears arcs assumed base bisect Book called centre circle circumference coincide common compounded const constructed contained definition demonstration described diameter difference divided double draw drawn equal equal angles equiangular equilateral equimultiples evident expressed external extremities fall figure follows four fourth geometry given given line given point greater half Hence homologous included inscribed intersect joining less line A B magnitudes manner mean meet multiple opposite parallel parallelogram pass perpendicular placed plane position problem produced proportional PROPOSITION proved pyramid radius ratio rectangle remaining respectively right angles right line segments sides sides A B similar solid solution square surface taken tangent THEOREM third triangle twice vertex vertices volume whole

### Popular passages

Page 16 - If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal.

Page 24 - 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 22 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.

Page 104 - ... be equimultiples, the one of the second, and the other of the fourth. Let A the first be the same multiple of B the second, that C the third is of D the fourth ; and of A, C let equimultiples EF, GH be taken.

Page 107 - ... If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; then if the first magnitude be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less.

Page 107 - N ; and if equal, equal ; and if less, less : but if G be greater than L, it has been shown that L HC K E M F N H is greater than M ; and if equal, equal; and if less, less: therefore, if G be greater than L, K is greater than N ; and if equal, equal ; and if less less : and G, K are any equimultiples whatever of A, E ; and L, N any whatever of B, F : therefore as A is to B, so is E to F (5.

Page 187 - If two angles of one triangle are equal to two angles of another triangle, the third angles are equal, and the triangles are mutually equiangular.

Page 107 - IF the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth...

Page 107 - THEOR. IF the first be the same multiple of the second, or the same part of it, that the third is of the fourth ; the first is to the second, as the third is to the fourth...

Page 107 - D (as in fig. 2 and 3), this magnitude can be multiplied, so as to become greater than D, whether it be AC, or CB. Let it be multiplied until it...