A treatise on problems of maxima and minima, solved by algebra. Repr. under the superintendence of A. De Morgan |
Common terms and phrases
a² a² a² b2 a²b a²b² a²x a²x² a²y² algebra altitude ax x² ax² ax³ ay² b(b² base bx² circumference cone consequently constant given quantity cos² cubic equation cylinder Delhi diameter easily be solved Ellipse equa equation we find equation x² evidently a minimum exactly divide find x find x² frustrum greater Hence it appears Hindoo impossible roots INSCRIBE THE GREATEST isosceles Let ABC m² n² mathematics Maxima and Minima maximum negative quantity negative roots pa² parabola perpendicular plane PROB px² quadratic equation quadratic we find radius Ramchundra rectangle right angle solved without impossible Solving this quadratic square surface taken so small variable x² sin²
Popular passages
Page 158 - The same may easily be solved without impossible roots. PROB. (8.) TO FIND THAT POINT WITHIN A GIVEN TRIANGLE, FROM WHICH IF LINES BE DRAWN TO THE ANGULAR POINTS, THE SUM OF THEIR SQUARES SHALL BE A MINIMUM. (Fig. 65.) Let ABC be the given triangle, and let BD — a, AC = b, AD = c, AE = x, EG =. у where G is the point required, .-. DE = x — с, FE — a — у, and EC = b — x, and therefore AG?+ CG...
Page 118 - QUESTION IV. It is required to determine the size of a ball, which, being let fall into a conical glass full of water, shall expel the most water possible from the glass ; its depth being 6, and diameter 5 inches.
Page 123 - PROB. (27.) TO SAW OUT OF THE TRUNK OF A TREE, A RECTANGULAR BEAM THAT SHALL HAVE THE GREATEST POSSIBLE POWER OF SUSPENSION. (Fig. 58.) Actual experiments lead to this result that in a parallelopipedon of uniform thickness, supported on two points and loaded in the middle, the lateral strength is directly as the product of the breadth into the square of the depth, and inversely as the length. Let ACBm be the circumference of the trunk and the rectangle AB the base or top of the beam cut out of the...
Page 3 - RULE.* Multiply the base by the perpendicular height, and half the product will be the area.
Page 4 - DC. That is. If the measure of the supplemental chord of any arc be increased by the number 2, the square root of the sum will be the supplemental chord of half that arc. Now...
Page 6 - The areas or spaces of circles are to each other as the squares of their diameters or of their radii.
Page ix - That sound judgment which gives men well to know what is best for them, as well as that faculty of invention which leads to development of resources and to the increase of wealth and comfort, are both materially advanced, perhaps cannot rapidly be advanced without, a great taste for pure speculation among the general mass of the people, down to the lowest of those who can read and write.
Page 74 - This is the celebrated problem of the form of the cells of bees. Maraldi was the first who measured the angles of the faces of the terminating solid angle, and he found them to be 109° 28' and 70° 32
Page 5 - CD, and the halves of all the sides, or the half perimeter of the polygon. Now, conceive the number of sides of the polygon to be indefinitely increased ; then will its perimeter coincide with the circumference of the circle, and consequently the altitude CD will become equal to the radius, and the whole polygon equal to the circle. Consequently, the space of the circle, or of the polygon in that state, is equal to the rectangle of the radius and half the circumference.
Page xxxii - This equation is called the equation of the Parabola, because it expresses the relation between the lines AP & Pm which determine the position of points on the curve. (3.) TO FIND THE EQUATION TO THE ELLIPSE. (Fig. 2.) Let two indefinite lines Sm, Hm, revolve, in a given plane, about the points S, H, and cut each other in m, in such a manner that Sm -f- mH may be an invariable quantity ; then the locus of the point m is an Ellipse.