PHYSICAL AND MATHEMATICAL SCIENCES. 31

straight or curved. For example, the locus of the ver-tices of triangles of equal area on the same base, is astraight line parallel to the base; and the locus of thevertices of right-angled triangles on the same hypothe-nuse, is a semicircle. These loci were chiefly employedas affording the means of solving other problems of thedeterminate kind.

Of these, one which largely occupied the attention ofthe ancient geometers, and received its solution in thePlatonic school, was a response of the oracle of Delos,requiring, that of the altar in that temple, which was anexact cube, the exact double in solid content should bemade also in the form of a cube. This was done atfirst mechanically ; but Menechme applied to the so-lution the resources of the method of loci, and produceda geometrical construction. The trisection of a circulararc was another problem of some celebrity, which, inlike manner, was made to yield to the growing powersof geometry, though it had resisted all attempts bymeans of the elementary methods.

To the principles developed in these several dis-coveries, we shall, at a future period, have occasion torecur. We will merely remark, in this place, that theyunquestionably contain the germs, as it were, of the mostvaluable inventions of modern times.

Upon the whole, the geometry of the ancient schoolsis that portion of their speculations to which we lookback with by far the greatest interest and satisfaction ;and to which we are really indebted for the removal ofthe difficulties which we should otherwise have to con-tend with, in limine, in all our enquiries. What were tothem matters of high and original discovery, now formthe necessary elements of all well-conducted education :and even though modern science had given us moredirect and easy methods of arriving at those resultswhich we want for their actual applications, yet, as asubject of abstract study, for the refined elegance oftheir train of deduction, for the exact taste of theirstyle, and the fastidious precision of their reasoning,