## A Course of Mathematics: For the Use of Academies, as Well as Private Tuition, Volume 2 |

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Results 1-5 of 77

Page 3

Now, if from these equations it were required to find expressions for the angles of

a plane triangle, when the sides are given; we have only to

these equations by a, the second by b, the third by c, and to subtract each of the ...

Now, if from these equations it were required to find expressions for the angles of

a plane triangle, when the sides are given; we have only to

**multiply**the first ofthese equations by a, the second by b, the third by c, and to subtract each of the ...

Page 9

This indeed may always be done in an expression, however complex, by merely

rendering all the terms homogeneous; that is, by

power of R as shall make it of the same dimension, as the term in the equation ...

This indeed may always be done in an expression, however complex, by merely

rendering all the terms homogeneous; that is, by

**multiplying**each term by such apower of R as shall make it of the same dimension, as the term in the equation ...

Page 12

A',

second number by 2. Both these expressions for the sine of half an arc or angle

will be of use to us as we proceed. 21. If the values of sin. (A + B) and sin. (A – B),

given ...

A',

**multiplying**the quantities under the radical by 4, and dividing the wholesecond number by 2. Both these expressions for the sine of half an arc or angle

will be of use to us as we proceed. 21. If the values of sin. (A + B) and sin. (A – B),

given ...

Page 15

... (putting cos B. cos c-sin B : sin c for cos (b+c)), is=sin A. cos s. cos c-sin A. sin B

: sin c-H cos A. sin (b+c); and,

results sin A. sin c-Hsin B. sin (A+B+C) = sin A. cos B : cos c. sin B+sin A. sin c.

... (putting cos B. cos c-sin B : sin c for cos (b+c)), is=sin A. cos s. cos c-sin A. sin B

: sin c-H cos A. sin (b+c); and,

**multiplying**by sin B, and adding sin A. sinc, thereresults sin A. sin c-Hsin B. sin (A+B+C) = sin A. cos B : cos c. sin B+sin A. sin c.

Page 18

sum of the tangents % any three arcs which together con. stitute a circle,

. (XXX.) Since both arcs in the second and fourth quadrants have their tangents ...

sum of the tangents % any three arcs which together con. stitute a circle,

**multiplied**by the square of the radius, is equal to the product of those tangents. . .. (XXX.) Since both arcs in the second and fourth quadrants have their tangents ...

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### Popular passages

Page 13 - 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.

Page 465 - Or, by art. 249 of the same, the pressure is equal to the weight of a column of the fluid...

Page 70 - To prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles (see fig.

Page 295 - The workmen thought that substituting part silver was only a proper <perquisite; which taking air, Archimedes was appointed to examine it ; who, on putting...

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 245 - BPC) ; or, the pressure of a fluid on any surface is equal to the weight of a column of the fluid...

Page 297 - In the doctrine of fluxions, magnitudes or quantities of all kinds are considered as not made up of a number of small parts, but as generated by continued motion, by means of which they increase or decrease ; as a line by the motion of a point ; a surface by the motion of a line ; and a solid by the motion of a surface.

Page 250 - Weigh the denser body and the compound mass, separately, both in water, and out of it ; then find how much each loses in water, by subtracting its weight in water from its weight in air; and subtract the less of these remainders from the greater. Then...

Page 490 - The reason is, all bodies lose some of their weight in a fluid, and the weight which a body loses in a fluid, is to its whole weight, as the specific gravity of the fluid is to that of the body.

Page 457 - ... horizontal *. 2. The theorems just given may serve to show, in what points of view machines ought to be considered by those who would labour beneficially for their improvement. The first object of the utility of machines consists in furnishing the means of giving to the moving force the most commodious direction ; and, when it can be done, of causing its action to be applied immediately to the body to be moved. These can rarely be united : but the former can be accomplished in most instances...