## The Elements of Euclid, Viz: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected; and Some of Euclid's Demonstrations are Restored. Also the Book of Euclid's Data, in Like Manner Corrected. the first six books, together with the eleventh and twelfth |

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Page 121

... above the second , is to the second , as the

, is to the fourth . 17th prop . book 5 . XVII . Convertendo , by conversion ; when

there are four proportionals , Book V. als , and it is inferred , that OF EUCLID .

... above the second , is to the second , as the

**excess**of the third above the fourth, is to the fourth . 17th prop . book 5 . XVII . Convertendo , by conversion ; when

there are four proportionals , Book V. als , and it is inferred , that OF EUCLID .

Page 122

Book V. als , and it is inferred , that the first is to its excefs above the second , as

the third to its

distantia ) , or ex aequo , from equality of distance ; when there is any number of ...

Book V. als , and it is inferred , that the first is to its excefs above the second , as

the third to its

**excess**above the fourth Prop . E. book 5 . XVIII . Ex aequali ( sc .distantia ) , or ex aequo , from equality of distance ; when there is any number of ...

Page 144

... proportionals by conversion , that is , the first is to its

as the third to its

BA is to AE , as DC to CF. C Because AB is to BE , as CD to DF , by divi- E fion ...

... proportionals by conversion , that is , the first is to its

**excess**above the second ,as the third to its

**excess**above the fourth , А Let AB be to BE , as CD to DF ; thenBA is to AE , as DC to CF. C Because AB is to BE , as CD to DF , by divi- E fion ...

Page 149

Q. E. D. COR . 1. If the same hypothesis be made as in the proposi . tion , the

the third and fixth to K3 the OF EUCLID . 149 Next, Let there be four magnitudes,

A, ...

Q. E. D. COR . 1. If the same hypothesis be made as in the proposi . tion , the

**excess**of the first and fifth fhall be to the second , as 2 22. 3 Book V. the**excess**ofthe third and fixth to K3 the OF EUCLID . 149 Next, Let there be four magnitudes,

A, ...

Page 150

Book V. the

is the same with that of the proposition , if division be used instead of composition

. Cor . 2. The proposition holds true of two ranks of magni . tudes , whatever be ...

Book V. the

**excess**of the third and fixth to the fourth : The demonftra . tion of thisis the same with that of the proposition , if division be used instead of composition

. Cor . 2. The proposition holds true of two ranks of magni . tudes , whatever be ...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe becauſe Book Book XI caſe circle circle ABCD circumference common cone cylinder definition demonſtrated deſcribed diameter divided double draw drawn equal equal angles equiangular equimultiples exceſs fame fides firſt folid fore four fourth given angle given in poſition given magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane produced prop proportionals propoſition pyramid radius rectangle rectangle contained remaining right angles ſame ſecond ſegment ſhall ſides ſimilar ſolid ſphere ſquare ſquare of AC Take taken theſe third triangle ABC wherefore whole

### Popular passages

Page 32 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...

Page 165 - 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.

Page 170 - 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 10 - When several angles are at one point B, any ' one of them is expressed by three letters, of which ' the letter that is at the vertex of the angle, that is, at ' the point in which the straight lines that contain the ' angle meet one another, is put between the other two ' letters, and one of these two is...

Page 55 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Page 32 - ... 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 45 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 211 - AB shall be at right angles to the plane CK. Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK ; take any point F in CE, from which draw FG in the plane DE at right D angles to CE ; and because AB is , perpendicular to the plane CK, therefore it is also perpendicular to every straight line in that plane meeting it (3.

Page 38 - 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 304 - Thus, if B be the extremity of the line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length : For if it have any, this length must either be part of the length of the line AB, or of the line KB.