The Works of Archimedes (Dover Books on Mathematics)
Archimedes
Language: English
Pages: 326
ISBN: 0486420841
Format: PDF / Kindle (mobi) / ePub
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HenceKG = 3GF ButKG = 3LK, from (1) above. Therefore And, from (2), LF = (AO−AL−OF)=AO = OF. ThereforeOF = 5GF, andOG = 6GF. ButAO = 3OF = 15GF. Therefore, by subtraction, Proposition 9 (Lemma). If a, b, c, d be four lines in continued proportion and in descending order of magnitude, and if d : (a−d) = x : (a −c), and (2a + 4b + 6c + 3d) : (5a +10b + 10c + 5d) = y : (a − c), is required to prove that x+y = a. [The following is the proof given by Archimedes, with the only difference
by the use of this same lemma that they have shown that circles are to one another in the duplicate ratio of their diameters, and that spheres are to one another in the triplicate ratio of their diameters, and further that every pyramid is one third part of the prism which has the same base with the pyramid and equal height; also, that every cone is one third part of the cylinder having the same base as the cone and equal height they proved by assuming a certain lemma similar to that aforesaid.
is a vertical straight line. “For this is proved †.” Therefore, as before, there will be equilibrium. Thus orP = ΔBCD. Proposition 8, 9. Suppose a lever AOB placed horizontally and supported at its middle point O. Let a triangle BCD, right-angled or obtuseangled at C, be suspended from the points B, E on OB, the angular point O being so attached to E that the side CD is in the same vertical line with E. Let Q be an area such that SO : OE = ΔBCD : Q. Then, if ctu area P suspended from A
is known. Therefore BO2, or OE2, can be found, and therefore O. * To prove this, suppose that, in the figure on the opposite page, BR1 is produced to meet the outer parabola in R2. We have, as before, whenceER1 : ER = BQ2 : BQ1. And, since R1 is a point within the outer parabola, ER : ER1 = BR1 : BR2, in like manner. HenceBQ1 : BQ2 = BR1 : BR2. BOOK OF LEMMAS. Proposition 1. If two circles touch at A, and if BD, EF be parallel diameters in them, ADF is a straight line [The proof in the
is no special technical term, but we find such phrases as the following: άν εις τòτ κύκλον εθεîα γραµµή �πéση if in a circle a straight line be placed, and the chord is then the straight line so placed ή �πεσοσα , or quite commonly ή ν τ κύκλω (εθεîα) simply. For the chord subtending one 656th part of the circumference of a circle we have the following interesting phrase, ποτεìνονσα éν τµµα διαιρεθείσας τς το ΑΒΓ κύκλου περιφερείας éς χνς’. A segment of a circle is τµµα κύκλου ; sometimes, to