## The Elements of Euclid: 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. viz. the first six books, together with the eleventh and twelfthMathew Carey, and sold by J. Conrad & Company, S. F. Bradford, Birch & Small, and Samuel Etheridge. Printed by T. & G. Palmer, 116, High-Street., 1806 - Trigonometry - 518 pages |

### From inside the book

Results 6-10 of 10

Page 272

Wherefore , every cone , & c .

cylinders of the same altitude are to one another as their bases . Let the cones

and cylinders , of which the bases are the circles ABCD , EFGH , and the axes KL

...

Wherefore , every cone , & c .

**Q. E. D. PROP**. XI . THEOR . See N. CONES andcylinders of the same altitude are to one another as their bases . Let the cones

and cylinders , of which the bases are the circles ABCD , EFGH , and the axes KL

...

Page 281

R. E. D.

one another as their altitudes . Let the ... Wherefore , cones , & c .

B D M M B. XII .

cylinder PG; ...

R. E. D.

**PROP**. XIV . THEOR . CONES and cylinders upon equal bases are toone another as their altitudes . Let the ... Wherefore , cones , & c .

**Q. E. D.**B D M M B. XII .

**PROP**. XV . THEOR . See 2 N OF EUCLID . 281 axis KL andcylinder PG; ...

Page 494

cies AB , AC passing through its pole A in B , C , the angle at the centre of the

sphere upon the circumference BČ , is the same with the spherical angle BAC ,

and ...

**Q. E. D. PROP**. III . FIG . 2 . IF a great circle be described meeting two great cir .cies AB , AC passing through its pole A in B , C , the angle at the centre of the

sphere upon the circumference BČ , is the same with the spherical angle BAC ,

and ...

Page 500

Q. E. D. Case 3. Let AB be greater than a quadrant , make AE a quadrant , and let

a great circle pass through C , E , ECA is a right angle as before , and BCA is

greater than ECA , that is , greater than a right angle .

Q. E. D. Case 3. Let AB be greater than a quadrant , make AE a quadrant , and let

a great circle pass through C , E , ECA is a right angle as before , and BCA is

greater than ECA , that is , greater than a right angle .

**Q. E. D. PROP**. XIV . Page 511

the sides BC , CD , are proportional to the co - sines of the bases BA , AD . For by

21. the co - sine of BC is to the co - sine of BA , as ( the co - sine of AC to the ...

**Q. E. D. PROP**. XXV . FIG . 17 , 18 . THE same things remaining , the co - sines ofthe sides BC , CD , are proportional to the co - sines of the bases BA , AD . For by

21. the co - sine of BC is to the co - sine of BA , as ( the co - sine of AC to the ...

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### Common terms and phrases

ABCD added altitude angle ABC angle BAC base Book centre circle circle ABC circumference common cone cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples excess fore four fourth given angle given in position given in species given magnitude given ratio given straight line greater Greek half join less likewise magnitude manner meet multiple opposite parallel parallelogram pass perpendicular plane prisms produced PROP proportionals proposition pyramid Q. E. D. PROP radius reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid solid angle sphere square square of AC taken THEOR third triangle ABC wherefore whole

### Popular passages

Page 30 - Any two sides of a triangle are together greater than the third side.

Page 64 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.

Page 30 - IF, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. Let...

Page 59 - PROP. VIII. THEOR. IF a straight line be divided into any two parts, tour times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 28 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Page 165 - 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 19 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.

Page 191 - In right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar, and similarly described figures upon the sides containing the right angle.

Page 39 - 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 sidef. For any rectilineal figure ABCDE can 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 180 - Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.