CfiAr. VIII.
GEOMETRY.
779
Grecian schools to have a thorough and clear notion of form before an object could be rightlycomprehended, it is equally so at the present day : let us remember what the A
artizans of Athens have left us, and then seek for the rules by which theywere guided, and have before us the precepts of Plato in all our inquiries:that great philosopher in his “ Phasdo” observes : “ If we are not able to findout truth, this must be owing to one of two reasons; either that there is no itruth in the nature of things themselves, or that the mind of man is, from Isome radical defect, unable to discover it; upon the latter supposition, theinconstancy and uncertainty of human opinions can easily be accountedfor, and therefore we ought to ascribe all our errors to the defectiveness
of our own minds, and not to affirm, gratuitously, thatthere is any defect in the nature of things : truth is fre-quently difficult of access; therefore, before we arrive at it,we must proceed with great caution, examining carefullyevery step we take; and after all our efforts we shall oftenfind ourselves disappointed, and forced to confess our igno-rance.”
The Dodecaedron is formed round a pentagon in a similarmanner, as C A.
The dodecaedron may have its angles so cut awayas to produce a solid having twelve pentagons, united bytwenty hexagons, and having altogether thirty-two sides;in this shape it nearly approaches that of the globe. Arepresents the solid resting on one of its angles.
To draw the dodecaedron, describe first a circle of the dimen-sions to be given it, and divide it into ten equal parts, as shown atD E F G K I II L M N : these are to be united by lines, as shownin the figure; then from five alternate points, as E, G, I, L, N, drawother lines to the centre; then set out PQRST to show thepentagon at top, and omit the dotted lines to render the figurecomplete.
To construct the dodecaedron in card, draw first the pentagonT of the size or dimensions of the face; then on each of its sidesconstruct the other pentagons V, X, Y, Z, &c. of the same size;then, after cutting through the lines partly by which the latter
Fig.8G2.
Fig. 863.
are united to T, they may be folded and united together to resemble half the solid, asshown at A ; a similar process must be adopted for the other half, and then the two unitedto complete the dodecaedron.
The Icosaedron is formed by drawing twenty equal and equilateral triangles, and unitingthem in a manner to form the figure Q, A, and B.
This figure is formed oftwenty pyramids, whose basesare the twenty equal andequilateral triangles, the sum-mits of which terminate inthe centre of the body, asshown at A ; this figure re-presents one half of the ico-saedron, and the figure Bthe other. To draw thisform, trace a circle of thedimensions to be given to it,
and divide it into six equal parts, as shown atCDEFGH, then draw the sides CD, DE, E F,
F G, G H, and H C, to complete the hexagon, inwhich is then to be inscribed the equilateraltriangle HDF, and the parallelogram HDEG:then, from the point L, the centre of the sideH D, raise the perpendicular LC, and from the samepoint L, draw the right lines LI, LK, to the pointsI and K, which are the points where the two sidesH F and D F of the equilateral triangle HDF cutthe side G E of the parallelogram HDEG.
To construct it on card, twenty equal and equi-lateral triangles must be placed as shown at M N,and then, when the lines are partly cut through, theymay be folded, as shown at Q.
Fig 865.
. 806.
Fig.867.
Fig. 869.