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The first page of the
Book of Lemmas as seen in
The
Works of Archimedes (1897).
The Book of Lemmas is a book
attributed to Archimedes by Thābit ibn
Qurra. The book was written over 2,200 years ago and consists
of fifteen propositions on circles.^{[1]}
History
Translations
The Book of Lemmas was first introduced in Arabic by
Thābit ibn Qurra; he attributed the work to Archimedes. In 1661,
the Arabic manuscript was translated into Latin by Abraham Ecchellensis and edited by
Giovanni A. Borelli. The Latin
version was published under the name Liber
Assumptorum.^{[2]} T. L. Heath translated
Heiburg's Latin work into English in his The Works of
Archimedes.^{[3]}^{[4]}
Authorship
The original authorship of the Book of Lemmas has been
in question due to the fact that in proposition four, the book
refers to Archimedes in third person; however, it has been
suggested that it may have been added by the translator.^{[5]} Another
possibility is that the Book of Lemmas may be a collection
of propositions by Archimedes later collected by a Greek
writer.^{[6]}
New
geometrical figures
The Book of Lemmas introduces several new geometrical figures.
Arbelos
The arbelos is the shaded region (grey).
Archimedes' first introduced the arbelos in proposition four of
his book:
“ |
If AB be the diameter of
a semicircle and N any point on AB, and if semicircles be described
within the first semicircle and having AN, BN as diameters
respectively, the figure included between the circumferences of the
three semicircles is "what Archimedes called αρβηλος"; and its area
is equal to the circle on PN as diameter, where PN is perpendicular
to AB and meets the original semicircle in P.^{[6]} |
” |
The figure is used in propositions four through eight. In
propositions five, Archimedes introduces the Archimedes' twin circles, and
in proposition eight, he makes use what would be the Pappus chain,
formally introduced by Pappus of Alexandria.
Salinon
The salinon is the blue shaded region.
Archimedes' first introduced the salinon in proposition fourteen
of his book:
“ |
Let ACB be a semicircle
on AB as diameter, and let AD, BE be equal lengths measured along
AB from A, B respectively. On AD, BE as diameters describe
semicircles on the side towards C, and on DE as diameter a
semicircle on the opposite side. Let the perpendicular to AB
through O, the centre of the first semicircle, meet the opposite
semicircles in C, F respectively. Then shall the area of the figure
bounded by the circumferences of all the semicircles be equal to
the area of the circle on CF as diameter.^{[6]} |
” |
Archimedes proved that the salinon and the circle are equal in
area.
Propositions
- If two circles touch at A, and if CD, EF be parallel diameters
in them, ADF is a straight line.
- Let AB be the diameter of a semicircle, and let the tangents to
it at B and at any other point D on it meet in T. If now DE be
drawn perpendicular to AB, and if AT, DE meet in F, then
DF = FE.
- Let P be any point on a segment of a circle whose base is AB,
and let PN be perpendicular to AB. Take D on AB so that
AN = ND. If now PQ be an arc equal to the arc PA, and BQ
be joined, then BQ, BD shall be equal.
- If AB be the diameter of a semicircle and N any point on AB,
and if semicircles be described within the first semicircle and
having AN, BN as diameters respectively, the figure included
between the circumferences of the three semicircles is "what
Archimedes called αρβηλος"; and its area is equal to the circle on
PN as diameter, where PN is perpendicular to AB and meets the
original semicircle in P.
- Let AB be the diameter of a semicircle, C any point on AB, and
CD perpendicular to it, and let semicircles be described within the
first semicircle and having AC, CB as diameters. Then if two
circles be drawn touching CD on different sides and each touching
two of the semicircles, the circles so drawn will be equal.
- Let AB, the diameter of a semicircle, be divided at C so that
AC = 3/2 × CB [or in any ratio]. Describe
semicircles within the first semicircle and on AC, CB as diameters,
and suppose a circle drawn touching the all three semicircles. If
GH be the diameter of this circle, to find relation between GH and
AB.
- If circles are circumscribed about and inscribed in a square,
the circumscribed circle is double of the inscribed square.
- If AB be any chord of a circle whose centre is O, and if AB be
produced to C so that BC is equal to the radius; if further CO
meets the circle in D and be produced to meet the circle the second
time in E, the arc AE will be equal to three times the arc BD.
- If in a circle two chords AB, CD which do not pass through the
centre intersect at right angles, then (arc AD) + (arc
CB) = (arc AC) + (arc DB).
- Suppose that TA, TB are two tangents to a circle, while TC cuts
it. Let BD be the chord through B parallel to TC, and let AD meet
TC in E. Then, if EH be drawn perpendicular to BD, it will bisect
it in H.
- If two chords AB, CD in a circle intersect at right angles in a
point O, not being the centre, then
AO2 + BO2 + CO2 + DO2 = (diameter)2.
- If AB be the diameter of a semicircle, and TP, TQ the tangents
to it from any point T, and if AQ, BP be joined meeting in R, then
TR is perpendicular to AB.
- If a diameter AB of a circle meet any chord CD, not a diameter,
in E, and if AM, BN be drawn perpendicular to CD, then
CN = DM.
- Let ACB be a semicircle on AB as diameter, and let AD, BE be
equal lengths measured along AB from A, B respectively. On AD, BE
as diameters describe semicircles on the side towards C, and on DE
as diameter a semicircle on the opposite side. Let the
perpendicular to AB through O, the centre of the first semicircle,
meet the opposite semicircles in C, F respectively. Then shall the
area of the figure bounded by the circumferences of all the
semicircles be equal to the area of the circle on CF as
diameter.
- Let AB be the diameter of a circle., AC a side of an inscribed
regular pentagon, D the middle point of the arc AC. Join CD and
produce it to meet BA produced in E; join AC, DB meeting in F, and
Draw FM perpendicular to AB. Then EM = (radius of
circle).^{[6]}
References
- ^
Archimedes;
Heath, Thomas (1897), The works of Archimedes, University
Press
- ^
"From Euclid to Newton".
Brown University. http://www.brown.edu/Facilities/University_Library/exhibits/math/nofr.html. Retrieved
2008-06-24.
- ^
Aaboe, Asger (1997), Episodes from the Early
History of Mathematics, Washington, D.C.: Math. Assoc. of
America, pp. 77, 85, ISBN 0883856131, http://books.google.com/books?id=5wGzF0wPFYgC&printsec=frontcover, retrieved
2008-06-19
- ^
Glick,
Thomas F.; Livesey, Steven John; Wallis, Faith (2005), Medieval Science,
Technology, and Medicine: An Encyclopedia, New York:
Routledge, p. 41, ISBN 0415969301, http://books.google.com/books?id=SaJlbWK_-FcC&printsec=frontcover#PPT9,M1, retrieved
2008-06-19
- ^
Bogomolny, A. "Archimedes' Book of
Lemmas". Cut-the-Knot. http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/index.shtml. Retrieved
2008-06-19.
- ^ ^{a}
^{b}
^{c}
^{d}
Heath, Thomas Little
(1897), The Works of
Archimedes, Cambridge University, pp. xxxii,
301–318, http://books.google.com/books?id=bTEPAAAAIAAJ&printsec=titlepage, retrieved
2008-06-15