778
THEORY AND PRACTICE OF ENGINEERING.
Book II.
5
R
D
T
V
Q
Fig. 857.
Fig. 858.
of three plane triangles, the construction of such may be reduced to the considerationof the trihedral: as to the remaining surface, which encloses the solid, completelymaking a fourth side to the trihedral, it may be of any form whatever, regular, or irregular,or consisting of many surfaces ; it or they have nothing to do in the construction. Theportions of the trihedral, which may be obtained from three given parts, are the very sameas the three found in a sphericaltriangle from three given parts :this is in fact spherical trigono-metry.
This figure is easily compre-hended by the plan, on which, inthe form of the cross, the sixsquares that are to make the sidesare set out Q becomes the top,
Soneend, andP the other, RT thesides, and D the bottom. Apieceof card-board, partly cut through,where the lines are drawn, may befolded on the contrary side so asto exhibit this figure. It will ap-pear at once evident that each facehas one opposite to it as well asparallel, and that the oppositeedges are parallel; the straightline which joins two oppositeangles passes through the centreof the cube.
To draw and construct an Octaedron. — Trace thesquare N O P Q, of the given dimensions : on each sideconstruct an equilateral triangle, which being folded to-gether will form one half of the octaedron; this repeatedand joined to the first half constitutes the entire figure.
The equilateral triangle, the square, and the pentagon,are the only forms which enter into the regular poly-edrons, whose angles and sides are equal; the solidangles of all, when cut away, regularly form figures, alsosymmetrical. 'The angles of the tetraedron may be sotaken off, that we may obtain a polyedron of eight faces,composed of four hexagons and four equilateral triangles,forming a polyedron of fourteen faces.
When it is required to show a bird’s-eye view of theoctaedron, it may be drawn either in simple outline, orit may be shaded to express its figure. The squareCDEF, by its dotted diagonals, shows the plan of itssides; GHIKL its solid form, and M the figure withits sides tinted.
To construct the octaedron on card board, it is onlynecessary to cut partly through lines which unite thetriangles with the side of the square, and then to bringthem into the position shown at NOPQ, and unitingthe edges RR to form one half, as TR. Similar oper-ation for the other half V must then be performed, andthe two brought base to base, to complete the modelof the entire octaedron.
These regular solids have all been admirably cut by theaid of machinery, and are very useful in enabling us to com-prehend the structure of minerals, or to project the forminto which a stone requires to be shaped for particularsituations in masonry. In the academy of the Greeksconsideration was deservedly given to the five Platonicbodies, so designated after their great discoverer, whoalways taught by example or with models before him.
In France great attention has been paid to the subject ofprojection, and to the right understanding of the regularand irregular solids ; rules are laid down in every treatiseupon carpentry and masonry to enable the student to per-form these operations, and it is in proportion as he com-prehends form, that he can construct with strength and economy.
Fig. 859.
If it were necessarv in the