By David Perkins
Calculus & Its Origins is an summary of calculus as an highbrow pursuit having a 2,000-year history.
Author David Perkins examines the level to which mathematicians and students from Egypt, Persia, and India absorbed and nourished Greek geometry, and info how the students wove their inquiries right into a unified theory.
Chapters disguise the tale of Archimedes discovery of the realm of a parabolic phase; ibn Al-Haytham s calculation of the quantity of a revolved zone; Jyesthadeva s clarification of the countless sequence for sine and cosine; Wallis s deduction of the hyperlink among hyperbolas and logarithms; Newton s generalization of the binomial theorem; Leibniz s discovery of integration via parts--and a lot more.
Each bankruptcy additionally comprises workouts by way of such mathematical luminaries as Pascal, Maclaurin, Barrow, Cauchy, and Euler. Requiring just a uncomplicated wisdom of geometry and algebra--similar triangles, polynomials, factoring--and a willingness to regard the limitless as metaphor--Calculus & Its Origins is a treasure of the human mind, pearls strung jointly via mathematicians throughout cultures and centuries.
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7). You might start by finding the length of CF . 8. ABCD is a square with sides 1, and both AEF and DCF are straight lines. 4 Extending ibn al-Haytham’s sums of powers. 12). 1 for a few cases, and then draw a universal conclusion from our results. 1 to find the formula that sums the fifth powers: n X n6 n5 5n4 n2 k5 D C C : 6 2 12 12 kD1 (b) Inspect your work for a reason to believe that for any positive integer p > 5, n X kD1 kp D npC1 C (terms of lower degree) : pC1 ✐ ✐ ✐ ✐ ✐ ✐ “C&IO” — 2012/2/14 — 12:16 — page 29 — #43 ✐ ✐ 3 Curves Say you bounce a ball Have you ever noticed that Between the business of its going up and the business of its fall it hesitates?
Roberval proved that it was exactly three times. 4. ✐ The cycloid AC G tracks the movement of point A on the circle as it rolls from A to G. 4. 5 (a), focus on point C as it moves to C 0 , tracing part of a cycloid. 5 (b). Let L be the point on CP that mirrors the height of C 0 , so LC 0 is always parallel to P G. Points N and M lie on LC 0 so that N remains on the ghost circle and M stays directly above the rolling circle’s point of contact with the ground. Roberval noted that LN D M C 0 at all times; as LN sweeps through the semicircle CNP , segment M C 0 sweeps through part of the region under the cycloid C C 0 G.
His serious treatment of zero illustrates this point; his bold claim that zero divided by zero equals zero foreshadowed a debate that we shall return to in chapter 9. Brahmagupta devoted much attention to arithmetic sequences, which will play an important role in chapter 5. A sequence of numbers is arithmetic if the common difference between consecutive terms is constant. For example, the arithmetic sequence 5; 8; 11; 14; 17; 20 has common difference 3. Its initial term is 5 and its period (that is, the number of terms in the sequence) is 6.