## Newton's Principia, First Book, Sections I., II., III.: With Notes and Illustrations, and a Collection of Problems Principally Intended as Examples of Newton's MethodsMacmillan, 1878 - 292 pages |

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### Common terms and phrases

AC² accelerating effect action angular velocity apse arcs described bisects body describes body moves catenary central orbit centre of force centre of gravity centripetal force chord of curvature circle of curvature conic section constant curve cycloid diameter of curvature directrix distance ellipse epicycloid equiangular spiral extremities fixed point focus force tending given point hence hyperbola impulse indefinitely diminished intersect latus rectum Lemma lemniscate length magnitude major axis orbit described ordinate oscillation parabola parallel parallelograms particle pendulum perp perpendicular plane polygon position PROP proposition prove quantities radius of curvature ratio of equality right angles SCHOLIUM shew sides similar space described square straight line string subtense tangent triangle ultimate ratio uniform velocity vanishes varies inversely vertex

### Popular passages

Page v - Newton, by showing the extent to which they may be applied in the solution of problems ; he has also endeavoured to give assistance to the student who is engaged in the study of the higher branches of Mathematics, by representing in a geometrical form several of the processes employed in the Differential and Integral Calculus, and in the analytical investigations of Dynamics, FROST and WOLSTENHOLME.—k TREATISE ON SOLID GEOMETRY. ' By PERCIVAL FROST, MA, and the Rev.

Page 234 - The third, viz. that the squares of the periodic times are proportional to the cubes of the mean distances...

Page 238 - In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance...

Page 232 - ... time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the sesquiplicate ratio of the greater axis, equal to the ratio of the periodic time. QED COR. Therefore the periodic times in ellipses are the same as in circles whose diameters are equal to the greater axes of the ellipses.

Page 64 - The tangent\ of an arc is a straight line drawn from one extremity of the arc, perpendicular to the...

Page 39 - For as the parallelograms in the one are severally to the parallelograms in the other, so (by composition) is the sum of all in the one to the sum of all in the other; and so is the one figure to the other; because (by Lem. Ill) the former figure to the former sum, and the latter figure to the latter sum, are both in the ratio of equality. QED COR.

Page 128 - The path traced out by a point on the circumference of a circle which rolls...

Page 269 - The cubes of the mean distances of the planets from the sun are proportional to the squares of their times of revolution.

Page 253 - A particle is describing an ellipse under the action of a force tending to the focus ; if, on arriving at the extremity of the minor axis, the force has its law changed, so that it varies as the distance, the magnitude at that point remaining...

Page 74 - Consequently, in the beginning of the motion the forces are as the spaces described directly, and the squares of the times inversely. COR. 5. And the squares of the times are as the spaces described directly and the forces inversely.