Some History of the Calculus of the Trigonometric Functions V. Frederick Rickey West Point

A Theorem for Triskaidekaphobics The 13 th is more likely to occur on Friday than on any other day of the week. The Gregorian calendar has a 400 year cycle. 7 does not divide 12∙400. So the days are not equally likely.

A Theorem for Triskaidekaphobics The 13 th is more likely to occur on Friday than on any other day of the week. Saturday684 Sunday687 Monday685 Tuesday685 Wednesday687 Thursday684 Friday688

Reviel Netz Professor of Classics at Stanford The Works of Archimedes: Translation and Commentary An editor of The Archimedes Palimpsest

Archimedes (died 212 BCE)

Sphere and Cylinder, Prop 21 If in an even-sided and equilateral polygon is inscribed inside a circle, and the lines are draw through, joining the sides of the polygon (so that they are parallel to one – whichever – of the lines subtended by two sides of the polygon), all the joined lines have to the same diameter of the circle that ratio, which the line (subtending the sides, whose number is smaller by one, than half the sides) has to the side of the polygon.

Problem Mesopotamians created trig, 3 rd BCE Hipparchus constructed a table, 150 BCE Archimedes was killed in 212 BCE So who did this? Cardano, Kepler, Roberval

What is a sine ? The Greeks used chords The Arabs used half-chords NB: These are line segments, not numbers!

Etymology Chord in Arabic: –Jya Half-chord in Arabic: –jiba Arabic abbreviation: –jb Latin mistranslation: –Jaib –Sinus

Etymology Chord in Arabic: –Jya Half-chord in Arabic: –jiba Arabic abbreviation: –jb Latin mistranslation: –Jaib –Sinus

Isaac Newton Series for arcsine and sine in De analysi, 1669 Portrait: Kneller 1689

Newton: 1664, 1676 (Epistola prior)

Gottfried Wilhelm von Leibniz The sine series could be derived from the cosine series by term- by-term integration

The derivatives of the trigonometric functions are rather amazing when one thinks about it. Of all the possible outcomes, D sin x = cos x. Simply cos x, not Is it just luck on the part of mathematicians who derived trig and calculus? I assume trig was developed before calculus, why or how could the solution prove to be so simple? Luck. A Student Fl. 1988

Roger Cotes Sir Isaac Newton, speaking of Mr. Cotes, said “If he had lived we might have known something.”

The small variation of any arc of a circle is to the small variation of the sine of that arc, as the radius to the sine of the complement.

Euler about 1737, age 30 Painting by J. Brucker 1737 mezzotint by Sokolov Black below and above right eye Fluid around eye is infected “Eye will shrink and become a raisin” Ask your ophthalmologist Thanks to Florence Fasanelli

Euler’s Life Basel Petersburg I Berlin Petersburg II ____ 76

Euler’s Calculus Books 1748Introductio in analysin infinitorum Institutiones calculi differentialis Institutiones calculi integralis _____ 2982

Euler was prolific I Mathematics29 volumes II Mechanics, astronomy31 III Physics, misc.12 IVa Correspondence 8 IVb Manuscripts 7 87 One paper per fortnight, Half of all math-sci work,

Euler creates trig functions in 1739

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From the preface of the Introductio

Chapter 1: Functions A change of Ontology: Study functions not curves

VIII. Trig Functions

He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author... Eulogy by Nicolas Fuss, 1783

Sinus totus = 1 π is “clearly” irrational Value of π from de Lagny Note error in 113 th decimal place “scribam π” W. W. Rouse Ball discovered (1894) the use of π in W m Jones Arcs not angles Notation: sin. A. z

Read Euler, read Euler, he is our teacher in everything. Laplace as quoted by Libri, 1846

Joseph Fourier

Georg Cantor,

Euler, age painting by Darbes In Geneva Used glass pane, á la Leonardo

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