Elements of geometry and trigonometry . , DE, will beto each other as the angles ACB, DCE. Scholium. Conversely, if the arcs, AB, DE, are to eachother as two whole numbers, the angles ACB, DCE will be toeach other as the same whole numbers, and we &hall haveACB : DCE : : AB : DE. For the partial arcs. Am, ;7z?2, & \)x, xy, &:c., being equal, the partial angles AC???, rnQn,ifec. and jyCx, xCy, <&c. will also be equal. PROPOSITION XVir. THEOREM. Whatever be the ratio of two angles, they will always he to eachother as the arcs intercepted between their sides ; the arcs beingdescribed from
Elements of geometry and trigonometry . , DE, will beto each other as the angles ACB, DCE. Scholium. Conversely, if the arcs, AB, DE, are to eachother as two whole numbers, the angles ACB, DCE will be toeach other as the same whole numbers, and we &hall haveACB : DCE : : AB : DE. For the partial arcs. Am, ;7z?2, & \)x, xy, &:c., being equal, the partial angles AC???, rnQn,ifec. and jyCx, xCy, <&c. will also be equal. PROPOSITION XVir. THEOREM. Whatever be the ratio of two angles, they will always he to eachother as the arcs intercepted between their sides ; the arcs beingdescribed from the vertices of the angles as centres, with equalradii. Let ACB be the greater andACD the less angle. Let the less angle be placedon the greater. If the i)ropo-sition is not true, the angleACB w^ill be to the angle ACDas the arc AB is to an arcgreater or less than AD. Suppose this arc to be greater, andlet it be represented by AG ; we shall thus have, the angleACE : aiifde ACD : : arc AB : arc AO. Next conceive the arc. LOOK 111. S3 AB to be divided into equal parts, each of which is less thanDO ; there will be at least one point of division between D andO ; let 1 be that point ; and draw CI. The arcs AB, AI, will be10 each other as two whole numbers, and by the precedingtheorem, we shall have, the angle ACB : angle ACI : : arc AB: arc AI. Comparing these two proportions with each other,we see that the antecedents are the same : hence, the conse-({uents are proj)ortional (Book II. Prop. IV.) ; and thus we findthe angle ACD : angle ACI : : arc AO : arc AI. But the arcAO is gf-eater than the arc AI ; hence, if this proportion is true,the angle ACD must be greater than the angle ACI : on thecontrary, however, it is less ; hence the angle ACB cannot beto the angle ACD as the arc AB is to an arc greater tlian a process of reasoning entirely similar, it may be shownthatthe fourth term of the proportion cannot be less than AD ;hence it is AD itself; therefore we have Angle ACB
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
