Proof that the series exhibits log-log growth

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Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as log log n. The proof is due to Ivan Niven[1], adapted from the the product expansion idea of Euler. In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes.

The proof rests upon the following four inequalities:

  • Every positive integer i can be uniquely expressed as the product of a square-free integer and a square as a consequence of the fundamental theorem of arithmetic. Start with:

where the βs are 0 (the corresponding power of prime q is even) or 1 (the corresponding power of prime q is odd). Factor out one copy of all the primes whose β is 1, leaving a product of primes to even powers, itself a square. Relabeling:

where the first factor, a product of primes to the first power, is square free. Inverting all the is gives the inequality

To see this, note that

where

That is, is one of the summands in the expanded product A. And since is one of the summands of B, every i is represented in one of the terms of AB when multiplied out. The inequality follows.

e is transcendental

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Lemma 2.   for sufficiently large  .

Proof. Note that

 

where   and   are continuous functions of   for all  , so are bounded on the interval  . That is, there are constants   such that

 

So each of those integrals composing   is bounded, the worst case being

 

It is now possible to bound the sum   as well:

 

where   is a constant not depending on  . It follows that

 

finishing the proof of this lemma.

Choosing a value of   satisfying both lemmas leads to a non-zero integer ( ) added to a vanishingly small quantity ( ) being equal to zero, an impossibility. It follows that that the original assumption, that   can satisfy a polynomial equation with integer coefficients, is also impossible; that is,   is transcendental.

Tartaglia's Triangle

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Tartaglia's Triangle from General Trattato di Numeri et Misure, Part II, Book 2, p. 69v.

Tartaglia was proficient with binomial expansions and included many worked examples in Part II of the General Trattato, one a detailed explanation of how to calculate the summands of  , including the appropriate binomial coefficients.[2]

Tartaglia knew of "Pascal's Triangle" one hundred years before Pascal, as shown in this image from the General Trattato. His examples are numeric, but he thinks about it geometrically, the horizontal line   at the top of the triangle being broken into two segments   and  , where point   is the apex of the triangle. Binomial expansions amount to taking   for exponents   as you go down the triangle. The symbols along the outside represent powers at this early stage of algebraic notation:  , and so on. He writes explicitly about the additive formation rule, that (for example) the adjacent 15 and 20 in the fifth row add up to 35, which appears beneath them in the sixth row.[3]

Volume of a tetrahedron

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Tartaglia was a prodigious calculator and master of solid geometry. In Part IV of the General Trattato he shows by example how to calculate the height of a pyramid on a triangular base, that is, an irregular tetrahedron.[4]

The base of the pyramid is a   triangle  , with edges of length  , and   rising up to the apex   from points  ,  , and   respectively. Base triangle   partitions into   and   triangles by dropping the perpendicular from point   to side  . He proceeds to erect a triangle in the plane perpendicular to line   through the pyramid's apex, point  , calculating all three sides of this triangle and noting that its height is the height of the pyramid. At the last step, he applies what amounts to this formula for the height   of a triangle in terms of its sides   (the height from side   to its opposite vertex):

 

a formula deriving from the Law of Cosines (not that he cites any justification in this section of the General Trattato).

Tartaglia drops a digit early in the calculation, taking   as  , but his method is sound. The final (correct) answer is:

 

The volume of the pyramid is easily gotten after that (not that Tartaglia gives it):

 

Simon Stevin invented decimal fractions later in the sixteenth century, so the last figure would have been foreign to Tartaglia, who always used fractions. All the same, his approach is in some ways a modern one, suggesting by example an algorithm for calculating the height of most or all irregular tetrahedra, but (as usual for him) he gives no explicit formula.

Ballistics

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Nova scientia (1537) was Tartaglia's first published work, described by Matteo Valleriani as:

... one of the most fundamental works on mechanics of the Renaissance, indeed, the first to transform aspects of practical knowledge accumulated by the early modern artillerists into a theoretical and mathematical framework.[5]

Then dominant Aristotelian physics preferred categories like "heavy" and "natural" and "violent" to describe motion, generally eschewing mathematical explanations. Tartaglia brought mathematical models to the fore, "eviscerat[ing] Aristotelian terms of projectile movement" in the words Mary J. Henninger-Voss.[6] One of his findings was that the maximum range of a projectile was achieved by directing the cannon at a 45° angle to the horizon.

Tartaglia's model for a cannonball's flight was that it proceeded from the cannon in a straight line, then after a while started to arc towards the earth along a circular path, then finally dropped in another straight line directly towards the earth[7] — examine the image above to get a sense of this for different elevations of the gun. At the end of Book 2 of Nova scientia, Tartaglia proposes to find the length of that initial rectilinear path for a projectile fired at an elevation of 45°, engaging in a Euclidean-style argument, but one with numbers attached to line segments and areas, and eventually proceeds algebraically to find the desired quantity (procederemo per algebra in his words).[8]

Mary J. Henninger-Voss notes that "Tartaglia's work on military science had an enormous circulation throughout Europe", being a reference for common gunners into the eighteenth century, sometimes through unattributed translations. He influenced Galileo as well, who owned "richly annotated" copies of his works on ballistics as he set about solving the projectile problem once and for all.[9]


Tartaglia's biographer Arnoldo Masotti writes that:

At the age of about fourteen, he [Tartaglia] went to a Master Francesco to learn to write the alphabet; but by the time he reached “k,” he was no longer able to pay the teacher. “From that day,” he later wrote in a moving autobiographical sketch, “I never returned to a tutor, but continued to labor by myself over the works of dead men, accompanied only by the daughter of poverty that is called industry” (Quesiti, bk. VI, question 8).[10]

Tartaglia moved to Verona around 1517, then to Venice in 1534, a major European commercial hub and one of the great centers of the Italian renaissance at this time. Also relevant is Venice's place at the forefront of European printing culture in the sixteenth century, making early printed texts available even to poor scholars if sufficiently motivated or well-connected — Tartaglia knew of Archimedes' work on the quadrature of the parabola, for example, from Guarico's Latin edition of 1503, which he had found "in the hands of a sausage-seller in Verona in 1531" (in mano di un salzizaro in Verona, l'anno 1531 in his words).[11]

Tartaglia eked out a living teaching practical mathematics in abacus schools and earned a penny where he could:

This remarkable man [Tartaglia] was a self educated mathematics teacher who sold mathematical advice to gunners and architects, ten pennies one question, and had to litigate with his customers when they gave him a worn out cloak for his lectures on Euclid instead of the payment agreed on.[12]

Translations

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Archimedes' works began to be studied outside the universities in Tartaglia's day as exemplary of the notion that mathematics is the key to understanding physics, Federigo Commandino reflecting this notion when saying in 1558 that "with respect to geometry no one of sound mind could deny that Archimedes was some god".[13] Tartaglia published a 71-page Latin edition of Archimedes in 1543, Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi, containing Archimedes' works on the parabola, the circle, centers of gravity, and floating bodies. Guarico had published Latin editions of the first two in 1503, but the works on centers of gravity and floating bodies had not been published before. Tartaglia published Italian versions of some Archimedean texts later in life, his executor continuing to publish his translations after his death. Galileo probably learned of Archimedes' work through these widely disseminated editions. [14]


  • Clagett, Marshall (1982). "William of Moerbeke: Translator of Archimedes". Proceedings of the American Philosophical Society. 126 (5): 356–366..
  • Malet, Antoni (2012). "Euclid's Swan Song: Euclid's Elements in Early Modern Europe". In Olmos, Paula (ed.). Greek Science in the Long Run: Essays on the Greek Scientific Tradition (4th c. BCE-17th c. CE). Cambridge Scholars Publishing. pp. 205–234. ISBN 978-1-4438-3775-0..
  • Tartaglia, Niccolò (1543). Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi..
  • Tartaglia, Niccolò (1543). Euclide Megarense philosopho..
  1. ^ Niven, Ivan, "A Proof of the Divergence of Σ 1/p", The American Mathematical Monthly, Vol. 78, No. 3 (Mar. 1971), pp. 272-273. The half-page proof is expanded by William Dunham in Euler: The Master of Us All, pp. 74-76.
  2. ^ See Tartaglia, Niccolò. General Trattato di Numeri et Misure, Part II, Book 2, p. 51v for expanding  .
  3. ^ See Tartaglia, Niccolò. General Trattato di Numeri et Misure, Part II, Book 2, p. 72r for discussion of the additive rule in "Pascal's Triangle".
  4. ^ See Tartaglia, Niccolò. General Trattato di Numeri et Misure, Part IV, Book 2, p. 35r for the calculation of the height of a 13-14-15-20-18-16 pyramid.
  5. ^ See Valleriani, Matteo, Metallurgy, Ballistics and Epistemic Instruments: The Nova scientia of Nicolò Tartaglia, p. 1.
  6. ^ Henninger-Voss, Mary J., "How the 'New Science' of Cannons Shook up the Aristotelian Cosmos", Journal of the History of Ideas 63, 3 (July 2002), pp. 371-397. "eviscerated": p. 376.
  7. ^ See Valleriani, Matteo, Metallurgy, Ballistics and Epistemic Instruments: The Nova scientia of Nicolò Tartaglia, pp. 169-181.
  8. ^ See Valleriani, Matteo, Metallurgy, Ballistics and Epistemic Instruments: The Nova scientia of Nicolò Tartaglia, pp. 176-177.
  9. ^ See Henninger-Voss, Mary J., "How the 'New Science' of Cannons Shook up the Aristotelian Cosmos", Journal of the History of Ideas 63, 3 (July 2002), pp. 391-393 for discussion and quote.
  10. ^ Masotti, Arnoldo, Niccolò Tartaglia in the Dictionary of Scientific Biography.
  11. ^ See Tartaglia, Niccolò. General Trattato di Numeri et Misure, Part IV, Book 3, p. 43v for the sausage seller
  12. ^ Zilsel, Edgar, The Social Origins of Modern Science, p. 35.
  13. ^ Clagett, Marshall, "William of Moerbeke: Translator of Archimedes", pp. 356-366.
  14. ^ Henninger-Voss, Mary J., "'New Science' of Cannons", p. 392.