Elements of plane and solid free-hand geometrical drawing, with lettering; and some elements of geometrical ornamental design, including the principals of harmonic angular ratios, etcIn three parts ..for draughtsmen and artisans; and teachers and students of industrial and mechanical drawing . =2x2x2x45. \ 360 = x40.(360=5 x72. ( 72=2x2x2x9. \ 45= ( 40=5 x8. ( 8=2x2x2x1. \ 9= ( 5=5 xl. That is : First, 360 contains, as before seen, 2 thrice as a fac-tor; 3, twice and 5 once, with quotients, after successively di-viding out all these factors, of 45, 40, and 72, in the t
Elements of plane and solid free-hand geometrical drawing, with lettering; and some elements of geometrical ornamental design, including the principals of harmonic angular ratios, etcIn three parts ..for draughtsmen and artisans; and teachers and students of industrial and mechanical drawing . =2x2x2x45. \ 360 = x40.(360=5 x72. ( 72=2x2x2x9. \ 45= ( 40=5 x8. ( 8=2x2x2x1. \ 9= ( 5=5 xl. That is : First, 360 contains, as before seen, 2 thrice as a fac-tor; 3, twice and 5 once, with quotients, after successively di-viding out all these factors, of 45, 40, and 72, in the three casesrespectively. Second, these quotients contain the same factorsin like manner, as shown in the second group. Third, the likeis true again, taking the quotients of the second group in theorder seen in the third group, where the final quotients areeach, 1. In this curious result, we see again, exhibited in numbers, theprincijple of unlfoiinity. But the order in which the first two groups of quotients areused as the next dividends, is found as indicated by the paren- * The number 7 is also excluded generally in the formation of musicalratios, but is said to be employed in Chinese music, and it enters into thecomposition of some peculiar theoretical systems, not in actual use. 1. NUMEKICAL EXPRESSION OF THE ELEMENTARY IDEAS. 83 theses and the adjoining circle, by combining 1, 2, 3 in everypossible order taken in rotation, in the direction of the . again we have likewise the _pri?iciple of variety. 23. Ratio the principle of comhination.—Numbers can becompared in two ways, by difference and by ratio ; differencebeing obtained by subtracting one number from another; andratio, by dividing one by the other. Aristotle (as quoted byHay) defines harmony as the union of contrary principles (as those of uniformity and variety) having a ratio to eachother. In hea.\\i\ixi\ forms, 2^^Oj)ortions constitute harmony;and Yitruvius defines harmony as the coinmensuration of thev
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Keywords: ., bo, bookcentury1800, bookdecade1870, booksubjectmechanicaldrawing
