in which they run. For if the artist knows what is beneath the surface, he has no difficulty in representing it as transparent. He must be careful however not to perplex himself by lines too much multiplied, and take advantage of his being able to paint the lines with different colours, for the purposes of distinction; and he must also use a considerable address in throwing out such lines as would be of little use, and retaining such as will produce the effect of a picture, which should be well preserved in order to make the exhibition easily intelligible. If he should wish to make a drawing of minerals or crystals, this perspective would be well suited to the purpose. The point, however, on which the writer of this paper can speak with the greatest confidence is on the representation of machines and philosophical instruments; having been himself so much in the habit of practically applying to them the principles that have been detailed: and this he has exemplified in the plates. The correct exhibition of objects would be much facilitated by the use of this perspective, even in the hands of a person who is but little acquainted with the art of drawing; and the information given by such drawings is much more definite and precise than that obtained by the usual methods, and better fitted to direct a workman in execution.* The author has transcribed this interesting paper from the first volume of the Transactions of the Cambridge Philosophical Society. The method is peculiarly deserving of the attention, and, in many cases, the adoption, of mechanics and engi neers. Some useful exemplifications of isoperimetrical perspective are given in the 18th and 19th volumes of the Mechanic's Magazine; and while this sheet was going through the press, I learnt that Mr. Jopling is now printing a small treatise on the subject, for the use of mechanics and artificers. CHAPTER VIII. MENSURATION OF SUPERFICIES AND SOLIDS. SECTION I-Mensuration of Superficies. The following rules will serve to find the areas or superficial contents of the figures whose names respectively precede them. 1. Rectangle, Square, Rhombus, or Rhomboid. Multiply the base into the height, for the area. 2. Triangle. Half base into the height. Or, continual product of two sides, and half the natural sine of their inclined angle. Or, when three sides, as a B, a C, B c, are given, their half sum beings; then area [sx (s-A B) X (SAC) XS- BC)]. 3. Trapezium. Base into sum of the perpendiculars. 4. Trapezoid. Multiply half the sum of the parallel sides into the perpendicular distance between them. 5. Irregular Polygon. Divide it into trapeziums, or trapezoids, and triangles, and find their areas separately their sum is the area of the polygon. 6. Regular Polygon. Multiply the square of the side given into the proper "multiplier for areas," printed in the table in Prob. 15, Practical Geometry, the product will be the area. 7. Circle. diameter : circumf :: 113 : 355 or, diameter: circumf :: 1: 3.141593 circumf : diameter :: 1 : 318309 area diameter squared × 785398 area = 8. Circular arc. in the arc its length. circumf. Radius of the circle x 017453 x degrees 9. Circular sector. arc x radius=area. 10. Circular segment. Multiply the square of the radius by either half the difference of the arc (of the segment) and its sine, or by half their sum, according as the segment is less or greater than a semicircle: the product will be the area. 11. Parabola. of the product of base and height = area. 12. Ellipse. Transverse axis x conjugate axis × 785398 area. す 13. The side of a square whose area shall be equal to that of a given circle, is nearly of the diameter, or more nearly 3d, ord, or 10 d, or 14 d; each approximating more nearly than the former. 167 SECTION II.-Mensuration of Solids. 1. Prism. 1. Superficies. Multiply the perimeter of one end by the length or height of the solid; the product will be the surface of the sides. To this add the areas of the two ends : the sum is the whole surface. 2. Solidity or Capacity area of the base x the height. = Note. The same rules serve for the surface and capacity of a cylinder. 2. Pyramid, or Cone. 1. Surface X slant height. = 2. Capacity area of base × height. 3. Frustrum of Pyramid. 1. Surface meters of the two ends x slant height. perimeter of the base sum of the peri 2. Capacity. Add a diameter or a side of the greater base to one of the less; from the square of the sum subtract the product of the said two diameters or sides: multiply the remainder by a third of the height; and this last product by 785398 for circles, or by the proper multiplier for polygons; the last product will be the capacity. That is, capacity=[(D+d) - Dd] m h. 4. Sphere. 1. Surface diameter squared x 3.141593. 2. Capacity diameter cubed × ·5236 or circumference cubed x⚫016887. 5. Spheric segment. 1. Surface circumf. sphere x height of segment. 2 h) where d = diam. h=height. 5. Paraboloid. Capacity-half base x height. This is a figure produced by the rotation of a parabola upon its axis. 6. Spheroid. This is a solid generated by the revolution of an ellipse about one of its axes. To find its capacity multiply the square of the revolving axis by the fixed axis, and that product by 5236. 7. Regular or platonic bodies, are comprehended by like, equal, and regular plane figures, and whose solid angles are all equal. There are only five regular solids, viz. The tetraedron, or regular triangular pyramid, having 4 triangular faces; The hexaedron, or cube, having 6 square faces; The dodacaedron, having 12 pentagonal faces; The icosaedron, having 20 triangular faces. PROB. 1. To find either the surface or the solid content of any of the regular bodies.-Multiply the proper tabular area or surface (taken from the following table) by the square of the linear edge of the solid for the superficies. And Multiply the tabular solidity in the last column of the table by the cube of the linear edge for the solid content. Surfaces and Solidities of regular Bodies, the side being unity or 1. L 2. The diameter of a sphere being given to find the side of any of the platonic bodies, that may be either inscribed in the sphere, or circumscribed about the sphere, or that is equal to the sphere. Multiply the given diameter of the sphere by the proper or corresponding number, in the following table, answering to the thing sought, and the product will be the side of the platonic body required. 27 $ 2 L 3. The side of any of the five platonic bodies being given, to find the diameter of a sphere, that may either be inscribed in that body, or circumscribed about it, or that is equal to it. As the respective number in the table above, under the title inscribed, circumscribed, or equal, is to 1, so is the side of the given platonic body to the diameter of its inscribed, circumscribed, or equal sphere. 4. The side of any one of the five platonic bodies being given to find the side of the other four bodies, that may be equal in solidity to that of the given body. As the number under the title equal in the last column of the table above, against the given platonic body, is to the number under the same title, against the body whose side is sought, so is the side of the given platonic body to the side of the body sought. Besides these there are thirteen demiregular bodies, called Solids of Archimedes. They are described in the Supplement to Lidonne's Tables de Tous les Diviseurs des Nombres, &c. Paris, 1808; twelve of them were described by Abraham Sharp, in his Treatise on Polyedra. SECTION III-Approximate Rules. 1. When the area of a field is known in square yards, to reduce them to acres, instead of dividing by 4840, multiply by 0002, which is much easier. Thus, suppose the area is found to be 56870 yards. Then 56870 |