User:Robert.Baruch/Arithmetica infinitorum

Dedication
To that most esteemed and celebrated man, most expert in mathematics, Doctor William Oughtred, rector of the church of Albury in Surrey.

Greetings.

Behold at long last that completed work of which the Cyclometric Proposition made hope, and which, forged by the printing house during the Passover holiday, I have dedicated to you (most celebrated man). And when, as is custom, it comes forth in public, it may be inscribed for someone, I thought of seeking no greater a man than a great mathematician to whom I should inscribe the work. And I perceived so easily that it should be to no one more than to you, who is both the most deserving of mathematicians and also whose writings I know with pleasure to have helped me. These writings, of course, are those you have bothjconcisely and clearly delivered in your Key to Mathematics, not a large work, if you will, writings which we have sought in vain for in many other volumes.

You will find this work (if I am any judge of it) wholly new. For indeed I see nothing I might call such, nor, I believe, shall others teach it. Although indeed it might not be doubted whether propositions known to others might be mixed in here and there,...

To be sure, this, our method, begins where it defines the method of Cavalieri called Indivisibles. From where a handle is given by that very work and very title. As indeed that one is called the Geometry of Indivisibles, thus I am led to call the method of mine the Arithmetic of Infinitesimals.

(Blah blah, lots of words)

I gave this at Oxon, July 19, 1655.

John Wallis.

The proposition which the preceding letter recounts is that which follows.

Yet another goddamn dedication
To that most esteemed man, Doctor William Oughtred, most celebrated of mathematicians in knowledge, John Wallis, Professor of Geometry at Oxon gives greetings.

...

The squaring of the circle
The equality-curve VC having been set forth, which the straight line VT meets in the corner, divided in however many equal parts, and just as many parallel lines led from each point of division up to the curve, of which the second is 1, the fourth 6, the sixth 30, the eighth 140, and so on. There will be, just as the second of them to the third, so too the semicircle to the square of the diameter.

Or, if the second be 1, the fourth 1 1/2, the sixth 1 7/8, and so on. There will be, just as the second to the third, so too the circle to the square of the diameter.

Or, if the second be 1, the fourth 2 1/2, the sixth 4 3/8, and so on. There will be, just as the second to the third, so too the triple of the circle to the quadruple of the squaring of the diameter.

I gave this from the Oxon Printing Press the day after Passover, 1655.

Proposition 1
Lemma. If a series of quantities continually increasing in arithmetic proportion is set forth (perhaps the natural sequence of numbers), beginning from the point, or 0 (zero, or null) (think like 0, 1, 2, 3, 4 and so on), let it be proposed to inquire how much has the ratio of the sum of all of them, to the sum of the same equal to the greatest.

Asides

 * aside-1.Blah blah blah