A Treatise on Plane and Spherical Trigonometry: Including the Construction of the Auxiliary Tables; a Concise Tract on the Conic Sections, and the Principles of Spherical Projection |
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Page 39
... join AF , DC ; and through E draw EG parallel to BC , meeting DC produced in G. Then it is evident that DB is the sum , and BE the difference , of AC and AB . The outward angle CAD of the triangle ABC , is equal to the two inward and ...
... join AF , DC ; and through E draw EG parallel to BC , meeting DC produced in G. Then it is evident that DB is the sum , and BE the difference , of AC and AB . The outward angle CAD of the triangle ABC , is equal to the two inward and ...
Page 40
... join DE and DC . Then , DAC being a right angle , DA : AC :: rad : tangent of ADC ( Art . 28 ) . Now , because AE AD , the angle ADE = AED ; hence each of those angles is half a right angle . Since the triangles ADE and ADC have the ...
... join DE and DC . Then , DAC being a right angle , DA : AC :: rad : tangent of ADC ( Art . 28 ) . Now , because AE AD , the angle ADE = AED ; hence each of those angles is half a right angle . Since the triangles ADE and ADC have the ...
Page 41
... join AE , FG ; bisect AB in H , and draw CD at right angles to AB . Then , since CD , which passes through the centre of the circle , cuts AG at right angles , E C AD = DG ( 3.3 ) ; or 6 D * SECTION I. 41 ADE and ADC have the angle at A ...
... join AE , FG ; bisect AB in H , and draw CD at right angles to AB . Then , since CD , which passes through the centre of the circle , cuts AG at right angles , E C AD = DG ( 3.3 ) ; or 6 D * SECTION I. 41 ADE and ADC have the angle at A ...
Page 46
... join BD , meet- OC in n ; through B , n , draw BH , nm , parallel to AO ; and np parallel to CF. = Then , since the arc BC CD , if we suppose BO and DO joined , those arcs will subtend equal angles at O ( 27.3 ) . Hence ( 4.1 ) Bn = Dn ...
... join BD , meet- OC in n ; through B , n , draw BH , nm , parallel to AO ; and np parallel to CF. = Then , since the arc BC CD , if we suppose BO and DO joined , those arcs will subtend equal angles at O ( 27.3 ) . Hence ( 4.1 ) Bn = Dn ...
Page 54
... Join DG , DB , CB ; and produce DB , CB , to meet AE in E and F. = Now , the angle ABC in a semicircle being a right one ( 31.3 ) , the adjacent angle ABF is also a right one ( 13.1 ) . Again , since AG and BG touch the circle , each of ...
... Join DG , DB , CB ; and produce DB , CB , to meet AE in E and F. = Now , the angle ABC in a semicircle being a right one ( 31.3 ) , the adjacent angle ABF is also a right one ( 13.1 ) . Again , since AG and BG touch the circle , each of ...
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Common terms and phrases
ABDP angled spherical triangle base bisect c.cos c.sin centre circle Art common section Comp AC cone conical surface consequently construction cosec cosine cotan directrix distance drawn EC² ecliptic ED² ellipse equal equation given angle greater axis Hence hyperbola hypothenuse join latus rectum less circle Let ABC line of measures logarithms meet opposite ordinate original circle parabola parallel perpendicular plane of projection primitive circle projected circle projected pole projecting point Q. E. D. ART Q. E. D. Cor quadrant radius right angled spherical right ascension right line secant semicircle semitangent sides similar triangles sine sphere spherical angle tangent tangent of half touches the circle triangle ABC vertex vertical angle whence wherefore
Popular passages
Page 32 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.
Page 39 - 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 95 - TO THEIR DIFFERENCE ; So IS THE TANGENT OF HALF THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE.
Page 98 - In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts ; or', to the rectangle under the cosines of the opposite parts.
Page 40 - Def. 10. 1.) If then CE is made radius, GE is the tangent of GCE, (Art. 84.) that is, the tangent of half the sum of the angles opposite to AB and AC. If from the greater of the two angles ACB and ABC, there be taken ACD their half sum ; the remaining angle ECB will be their half difference.
Page 36 - The side of a regular hexagon inscribed in a circle is equal to the radius of the circle.
Page 115 - The straight line joining the vertex and the centre of the base is called the axis of the cone.
Page 97 - The cotangent of half the sum of the angles at the base, Is to the tangent of half their difference...
Page 82 - If two angles of a spherical triangle are equal, the sides opposite these angles are equal and the triangle is isosceles. In the spherical triangle ABC, let the angle B equal the angle C. To prove that AC = AB. Proof. Let the A A'B'C
Page 82 - If two triangles have two angles of the one respectively equal to two angles of the other, the third angles are equal.