Higher Geometry and Trigonometry: Being the Third Part of a Series on Elementary and Higher Geometry, Trigonomentary and Mensuration : Containing Many Valuable Discoveries and Improvements in Mathematical Science, Especially in Relation to the Quadrature of the Circle, and Some Other Curves, as Well as the Cubature of Certain Curvilinear Solids : Designed as a Text-book for Collegiate and Academic Instruction, and as a Practical Compendium of Mensuration |
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Page 10
... draw the radii OA , OB , OC . Imagine the planes AOB , AOC , COB , to be drawn ; those planes will form a solid angle at the centre O ; and the angles AOB , AOC , COB , will be measured by AB , AC , BC , the sides of the spherical ...
... draw the radii OA , OB , OC . Imagine the planes AOB , AOC , COB , to be drawn ; those planes will form a solid angle at the centre O ; and the angles AOB , AOC , COB , will be measured by AB , AC , BC , the sides of the spherical ...
Page 13
... draw the in- definite arc MD perpendicular to AM ; take MD equal to a quadrant ; the point D will be one of the poles of the arc AM : or thus , at the two points A and M , draw the arcs AD and MD perpendicular to AM ; their point of ...
... draw the in- definite arc MD perpendicular to AM ; take MD equal to a quadrant ; the point D will be one of the poles of the arc AM : or thus , at the two points A and M , draw the arcs AD and MD perpendicular to AM ; their point of ...
Page 28
... draw two straight lines AD , BE , so as to cross each other at right angles , and from their point of inter- section , C , we discribe a circle with any radius so as to cut those lines in any points , as a , b , d , e , the ...
... draw two straight lines AD , BE , so as to cross each other at right angles , and from their point of inter- section , C , we discribe a circle with any radius so as to cut those lines in any points , as a , b , d , e , the ...
Page 30
... Draw the radius CP forming with CA any A angle ACP = 0 . From P draw PS perpendi- cular on CA. From A draw AT a tangent to the circumference at A. Produce CP to meet AT in T. T PB D S C E 4. Then the ratio of PS to the radius CA of ...
... Draw the radius CP forming with CA any A angle ACP = 0 . From P draw PS perpendi- cular on CA. From A draw AT a tangent to the circumference at A. Produce CP to meet AT in T. T PB D S C E 4. Then the ratio of PS to the radius CA of ...
Page 31
... Draw a circle A'B'D'E ' equal to the circle ABDE , and from C ' the centre , draw C'P ' , making with C'A ' the angle P'C'A ' equal to A ' the angle PCB ; that is , to the compliment of PCA , or to ( 90 ° -4 ) .. Then , since CP is ...
... Draw a circle A'B'D'E ' equal to the circle ABDE , and from C ' the centre , draw C'P ' , making with C'A ' the angle P'C'A ' equal to A ' the angle PCB ; that is , to the compliment of PCA , or to ( 90 ° -4 ) .. Then , since CP is ...
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abscissa altitude arithmetical progression axes base bisected chord circle circular circular segment circumference cone conjugate construction convex surface corresponding cosec cosine cylinder described diameter distance divided draw drawn ellipse equal to half equation expression feet formed formula frustum Geom geometrical given height hence hyperbola inches infinite series latus rectum length logarithm major axis multiplied opposite ordinates parabola parallel parallelogram passing perpendicular plane portion prism Prop PROPOSITION pyramid quadrant quadrature quantity radii radius ratio rectangle represent revoloidal surface right angles Scholium sector segment sides similar similar triangles sine solidity specific gravity sphere spherical triangle spheroid spindle square straight line tangent THEOREM tion triangle ABC Trigonometry ungula versed sine vertex vertical virtual centre whence
Popular passages
Page 81 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 81 - N .-. by definition, x — x" is the logarithm of ^ ; that is to say, The logarithm of a fraction, or of the quotient of two numbers, is equal to the logarithm of the numerator minus the logarithm of the denominator. III. Raise both members of equation (1) to the nth power. N"=a".
Page 68 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Page 7 - The radius of a sphere is a straight line, drawn from the centre to any point of the surface ; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface.
Page 138 - B . sin c = sin b . sin C cos a = cos b . cos c + sin b . sin c cos b = cos a . cos c + sin a . sin c cos A cos B cos c = cos a . cos b + sin a . sin b . cos C ..2), cotg b . sin c = cos G.
Page 8 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
Page 27 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees...
Page 78 - In a system of logarithms all numbers are considered as the powers of some one number, arbitrarily chosen, which is called the base of the system, and the exponent of that power of the base which is equal to any given number, is called the logarithm of that number. Thus, if a be the base of a system of logarithms, N any number, and x such that N = a* then x is called the logarithm of N in the system whose base is a.