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Extra resources for The Principia : Mathematical Principles of Natural Philosophy
Enable tangents hello and KL and issues B, C, and D accept. via any of the issues, B and D, draw an indefinite instantly line BD assembly the tangents in issues H and okay. Then, likewise, via any different issues, C and D, draw the indefinite immediately line CD assembly the tangents in issues I and L. lower BD in R and CD in S in any such means that HR could be to KR because the suggest proportional among BH and HD is to the suggest proportional among BK and KD and that's may be to LS because the suggest proportional among CI and identity is to the suggest proportional among CL and LD. And lower those traces at will both among issues okay and H, and among I and L, or outdoor them; then draw RS slicing the tangents in A and P, and A and P could be the issues of touch. For if A and P are meant to be the issues of touch positioned at any place at the tangents, and if via anyone of the issues H, I, ok, and L, say I, located in both tangent hello, the directly line IY is drawn parallel to the opposite tangent KL and assembly the curve in X and Y; and if during this line, IZ is taken with the intention to be the suggest proportional among IX and IY; then, from the Conics, the rectangle XI × IY or IZ2 could be to LP2 because the rectangle CI × identification to the rectangle CL × LD, that's (by construction), as SI2 to SL2, and therefore IZ can be to LP as SI to SL. for this reason the issues S, P, and Z lie in a single immediately line. moreover, because the tangents meet in G, the rectangle XI × IY or IZ2 might be to IA2 (from the Conics) as GP2 to GA2, and for that reason IZ may be to IA as GP to GA. hence, the issues P, Z, and A lie in a single directly line, and hence the issues S, P, and A are in a single directly line. And by means of an analogous argument it is going to be proved that the issues R, P, and A are in a single instantly line. for this reason the issues of touch A and P lie within the immediately line RS. And as soon as those issues were came upon, the trajectory might be defined as in prop. 23, case 1. Q. E. F. during this proposition and in prop. 23, case 2, the buildings are an identical even if the directly line XY cuts the trajectory in X and Y, and so they don't rely on this lower. yet as soon as the structures were proven for the case during which the instantly line does lower the trajectory, the buildings for the case within which it doesn't reduce the trajectory may be stumbled on; and for the sake of brevity i don't make the effort to illustrate them additional. Lemma 22 to alter figures into different figures of a similar type. permit or not it's required to transmute any determine HGI. Draw at will parallel instantly strains AO and BL slicing in A and B any 3rd line AB, given in place; and from any aspect G of the determine draw to the directly line AB the other instantly line GD parallel to OA. Then from a few element O, given in line OA, draw to the purpose D the immediately line OD assembly BL at d, and from the meeting-point erect the instantly line dg containing any given perspective with the instantly line BL and having a similar ratio to Od that DG has to OD; and g may be the element within the new determine hgi equivalent to element G.