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A Source Book in Mathematics, 1200-1800
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Historical SummaryGalileo GalileiAn application of Galilei’s theory of indivisibles is his derivation of the law for uniformly accelerated motion: if the acceleration is a, then and where the mean velocity between beginning and end. It will be seen that Galilei regards an area as generated by lines, or, we may say, as composed of lines—hence discarding the ancient difficulty that a sum of points can never be a line, and a sum of lines can never be an area. The text is again from the Dialogues concerning two new sciences, trans. H. Crew and A. de Salvio, 166–167; the original text is found in Opere, VIII, 208–209.The theorem had already appeared in scholastic writings (Selection III.1). P. Duhem, Etudes sur Léonard de Vinci (Hermarm, Paris, 1913), III. 388–398, called it "the rule of Oresme." See also C. B. Boyer, History of the calculus (Dover, New York, 1959), 83, 113; E. J. Dijksterhuis, The mechanization of the world picture (Clarendon Press, Oxford, 1961), 197–198; and A. Maier, An der Grenze von Scholastik und Naturwissenschaft (2nd ed.; Edizioni di Storia e Letteratura, Rome, 1953).From Theorem I, Proposition I, Galilei could pass without infinitesimals to Theorem II, Proposition II, which states that
Mathematics
4 GALILEI.
Accelerated Motion
THEOREM I, PROPOSITION I
The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed and the speed just before acceleration began.
Let us represent by the line AB [Fig. 1] the time in which the space CD is traversed by a body which starts from rest, at C and is uniformly accelerated; let the final and highest value of the speed gained during the interval AB be represented by the line EB drawn at right angles to AB; draw the line AE, then all lines drawn from equidistant points on AB and parallel to BE will represent
 Fig. 1
the increasing values of the speed, beginning with the instant A. Let the point F bisect the line EB; draw FG parallel to BA, and GA parallel to FB, thus forming a parallelogram AGFB which will be equal in area to the triangle AEB, since the side GF bisects the side AE at the point I; for if the parallel lines in the triangle AEB are extended to GI, then the sum of all the parallels contained in the quadrilateral is equal to the sum of those contained in the triangle AEB; for those in the triangle IEF are equal to those contained in the triangle GIA, while those included in the trapezium AIFB are common. Since each and every instant of time in the time interval AB has its corresponding point on the line AB, from which points parallels drawn in and limited by the triangle AEB represent the increasing values of the growing velocity, and since parallels contained within the rectangle represent the values of a speed which is not increasing, but constant, it appears, in like manner, that the momenta assumed by the moving body may also be represented, in the case of the accelerated motion, by the increasing parallels of the triangle AEB, and, in the case of the uniform motion, by the parallels of the rectangle GB. For what the momenta may lack in the first part of the accelerated motion (the deficiency of the momenta being represented by the parallels of the triangle AGI ) is made up by the momenta represented by the parallels of the triangle IEF.
Hence it is clear that equal spaces will be traversed in equal times by two bodies, one of which, starting from rest, moves with a uniform acceleration, while the momentum of the other, moving with uniform speed, is one-half its maximum momentum under accelerated motion. Q.E.D.
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Chicago: Galileo Galilei, "Accelerated Motion," A Source Book in Mathematics, 1200-1800, trans. A. De Salvio in A Source Book in Mathematics, 1200-1800, ed. D. J. Struik (Princeton: Princeton University Press, 1969, 1986), 208–209. Original Sources, accessed November 24, 2024, http://originalsources.com/Document.aspx?DocID=SXJNCGMMEM2SUJ6.
MLA: Galilei, Galileo. "Accelerated Motion." A Source Book in Mathematics, 1200-1800, translted by A. De Salvio, Vol. VIII, in A Source Book in Mathematics, 1200-1800, edited by D. J. Struik, Princeton, Princeton University Press, 1969, 1986, pp. 208–209. Original Sources. 24 Nov. 2024. http://originalsources.com/Document.aspx?DocID=SXJNCGMMEM2SUJ6.
Harvard: Galilei, G, 'Accelerated Motion' in A Source Book in Mathematics, 1200-1800, trans. . cited in 1969, 1986, A Source Book in Mathematics, 1200-1800, ed. , Princeton University Press, Princeton, pp.208–209. Original Sources, retrieved 24 November 2024, from http://originalsources.com/Document.aspx?DocID=SXJNCGMMEM2SUJ6.
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