off to work! i reviewed functions today and they make perfect sense now, more or less. tomorrow i’m reviewing a few more proofs then moving on to conic sections. totally exciting, i know! :) send positive vibes to me for work tonite plz. :)
This is one step on the way to connecting all conic sections together. It’s desperately needed, since the way they’re taught now is to have them be basically completely separate, and that’s not ok. This starts giving a sense of how you can start with a circle and start stretching it and get all the sections. But there are other ways to teach it too.
So my dear friends, I feel like this blog is going to be a mathblr for this week because that’s pretty much most of what I’ll be studying. I think I’ll post some maps i made for conic sections later. I hope you are having a blast! And I wish you good fortune in the ordeals of life.
Math Day2: more exercise questions and apple juice.
Apollonius contrives the ratios (for the parabola) PL:PA :: sq.BC:rect.BA,AC (for the hyperbola) PL:PP’ :: sq.AF:rect.BF,BC (for the ellipse) PL:PP’ :: sq.AF:rect.BF,FC in order to prove that (for the parabola) sq.QV = rect.LP,PV (for the hyperbola) sq.QV = RV,VP (for the ellipse) sq.QV = PV,VR which equate constant areas in a similar way as the mean proportional in a circle does. but how did Apollonius discover PL?
from point P in the three diagrams draw a line through the cone parallel to the base diameter BC. we’ll call it PS.
for a parabola: given Apollonius’ definition of the parameter or upright side: PL:PA :: sq.BC:rect.BA,AC, compounding the sides, PL:PA :: BC:BA comp. BC:AC, corresponding sides of similar figures being proportional PL:PA :: PS:PA comp. BM:PM, converted to a proportion of lines and areas, PL:PA :: rect.PS,BM:rect.PA,PM, the terms of the antecedent ratio raised to the common height PM, rect.PL,PM:rect.PA,PM :: rect.PS,BM:rect.PA,PM, but areas in the same ratio to the same are equal, so rect.PL,PM = rect.PS,BM, and since sides of equal rectangles are reciprocally proportional PL:PS :: BM:PM
for an hyperbola: given the definition of the parameter PL:PP’ :: sq.AF:rect.BF,BC, for compounding the sides PL:PP’ :: AF:BF comp. AF:BC, corresponding sides of similar figures being proportional PL:PP’ :: BM:MP comp. PS:PP’, decompounding PL:PP’ :: rect.BM,PS:rect.MP,PP’, raising PL and PP’ to common height MP, rect.PL,MP:rect.PP’,MP :: rect.BM,PS:rect.MP,PP’, things in the same proportion to the same are equal, so rect.PL,MP = rect.BM,PS, thus PL:PS :: BM:MP
for an ellipse: given the definition of the parameter PL:PP’ :: sq.AF:rect.BF,FC, compounding the sides PL:PP’ :: AF:BF comp. AF:FC, again substituting equal ratios for compounded ones, PL:PP’ :: BM:MP comp. PS:PP’, through similar triangles, decompounding thus PL:PP’ :: rect.BM,PS:rect.MP:PP’, raising PL:PP’ to similar height MP rect.PL,MP:rect.PP’:MP :: rect.BM,PS:rect.MP:PP’, again in the same ratio to equals we find rect.PL,MP = rect. BM,PS, therefore PL:PS :: BM:MP
so for all conic sections the parameter is defined by the same ratio, the parameter is always the line which is parallel to the base of the axial triangle drawn in the triangle at the vertex of the section, as the line, from the vertex side of the base to the point of intersection of the cutting plane and the base plane is to the line from the base to the vertex of the section along the common section of the axial triangle and the cutting plane; i.e. in the parabola and the hyperbola, the abcissa.
so why would Apollonius purposefully choose a complex definition that makes no sense without this much simpler one?