A Course of Mathematics for the Use of Academies, as Well as Private TuitionS. Campbell & son, E. Duyckinck, 1822 |
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Page 20
... Hence , substituting 1 for R and 3 , for cos a , in the expres- sion sin 2R2 ± 2R COS A ( equa . xII ) , it becomes sin 15 ° √2 - √✓✓ 3 = · 2588190 . Hence , sin 75 ° = cos 15 ° √ / 1−1 ( 2 − √√ / 3 ) = { √ // 2 + 1 / 3 = √6 ...
... Hence , substituting 1 for R and 3 , for cos a , in the expres- sion sin 2R2 ± 2R COS A ( equa . xII ) , it becomes sin 15 ° √2 - √✓✓ 3 = · 2588190 . Hence , sin 75 ° = cos 15 ° √ / 1−1 ( 2 − √√ / 3 ) = { √ // 2 + 1 / 3 = √6 ...
Page 23
... Hence the process is this : From tan Atan B + tan c≈ 5 · 3047057 Take tan A + tan B 3 : 1601988 Remains tan c 2.1445069 tan 65 ° From tan A + tan B + tan c А 5.3047057 Take tan B + tan c 3.8765577 1 : 4281480 tan 55 ° Remains tan a ...
... Hence the process is this : From tan Atan B + tan c≈ 5 · 3047057 Take tan A + tan B 3 : 1601988 Remains tan c 2.1445069 tan 65 ° From tan A + tan B + tan c А 5.3047057 Take tan B + tan c 3.8765577 1 : 4281480 tan 55 ° Remains tan a ...
Page 24
... hence the sides are 4 , 5 , 6 . The same conclusion is also readily obtained without the use of algebra . . 18 ° cos 72 ° is = 4R ( −1+ 1R ( 1 + √5 ) . Ex . 7. Demonstrate that sin √5 ) , and sin 54 ° cos 36 ° is Ex . 8. Demonstrate ...
... hence the sides are 4 , 5 , 6 . The same conclusion is also readily obtained without the use of algebra . . 18 ° cos 72 ° is = 4R ( −1+ 1R ( 1 + √5 ) . Ex . 7. Demonstrate that sin √5 ) , and sin 54 ° cos 36 ° is Ex . 8. Demonstrate ...
Page 27
... Hence it follows , that the surface of a spherical triangle BAC , and the three planes which determine it , form a kind of triangular pyramid , BCGA of which the vertex G is at the centre of the sphere , the base ABC a portion of the ...
... Hence it follows , that the surface of a spherical triangle BAC , and the three planes which determine it , form a kind of triangular pyramid , BCGA of which the vertex G is at the centre of the sphere , the base ABC a portion of the ...
Page 36
... Hence , the comparison of solid angles becomes a matter of great ease and simplicity ; for , since the areas of spherical triangles are measured by the excess of the sums of their an- gles each above two right angles ( th . 5 ) ; and ...
... Hence , the comparison of solid angles becomes a matter of great ease and simplicity ; for , since the areas of spherical triangles are measured by the excess of the sums of their an- gles each above two right angles ( th . 5 ) ; and ...
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Common terms and phrases
absciss altitude asymptotes axis ball beam body centre of gravity circle coefficient conic surface consequently Corol cosine Cotang cubic equation curve denote density descending determine diameter direction distance draw equa equal equation EXAM expression find the fluent fluid force given fluxion given plane gives greatest ground line Hence horizontal plane hyperbola inches inclined plane length logarithm maximum motion nearly negative ordinate parabola parallel perpendicular positive pressure prob PROBLEM produced proportional PROPOSITION quantity radius ratio rectangle resistance right angles right line roots Scholium side sine solid angle space specific gravity spherical triangle square straight line supposed surface Tang tangent theorem theref tion variable velocity vertical plane vertical projections vibrations weight whole α²
Popular passages
Page 437 - ... is equal to half the weight of a column of the fluid, whose base is equal to the surface pressed, and its altitude the same as that of the surface. Or, by art. 314 of the same, the pressure is equal to the weight of a column of the fluid...
Page 623 - NB In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's...
Page 154 - MECHANICAL POWERS are certain simple instruments employed in raising greater weights, or overcoming greater resistance than could be effected by the direct application of natural strength. They are usually accounted six in number; viz. the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw.
Page 240 - Then say, As the weight lost in water, Is to the whole weight, So is the specific gravity of water, To the specific gravity of the body.
Page 167 - The screw is a spiral thread or groove cut round a cylinder, and every where making the same angle with the length of it. So that if the surface of the cylinder, with this spiral thread on it, were unfolded and stretched into a plane, the spiral thread would form a straight inclined plane, whose length would be to its height, as the circumference of the cylinder...
Page 155 - A LEVER is any inflexible rod, bar, or beam, which serves to raise weights, while it is supported at a point by a fulcrum or prop, which is the centre of motion. The lever is supposed to be void of gravity or weight, to render the demonstrations easier and simpler. There are three kinds of levers.
Page 203 - The pressure of the fluid on any horizontal surface or plane, is equal to the weight of a column of the fluid, •whose base is equal to that plane, and altitude is its depth below the upper surface of the fluid.
Page 258 - ... preceding equation is only of the fourth power or degree ; but it is manifest that the above remark applies to equations of higher or lower dimensions : viz. that in general an equation of any degree whatever has as many roots as there are units in the exponent of the highest power of the unknown quantity, and...
Page 451 - ... increasing the charge, the velocity gradually diminishes, till the bore is quite full of powder. That this charge for the greatest velocity is greater as the gun is longer, but yet not greater in so high a proportion as the length of the gun is ; so that the part of the bore filled with powder, bears a less proportion to the whole bore in the long guns, than it does in the shorter ones ; the part which is filled being indeed nearly in the inverse ratio of the square root of the empty part.
Page 40 - In Every Spherical Triangle, the Sines of the Angles are Proportional to the Sines of their Opposite sides. If, from the first of the equations marked 1, the value of cos A be drawn, and substituted for it in the equation sin2 A = 1 — cos2 A, we shall have . , , cos2 a + cos