Thursday, 27 April 2023

Parabolas, Tangents, and the Wallace-Simson Line

Re-post from 2012, because of several visitors who ask questions that led me to refer them here.  Thought it worth re-posting.


The oft-called Simson line was attributed to Simson by Poncelet, but is now frequently known as the Wallace-Simson line since it does not actually appear in any work of Simson. (Oh go on, ask your teacher, so WHY do we still call it the Simson line at all?)
The Wallace for whom the line should more probably be named is William Wallace FRSE (23 September 1768, Dysart—28 April 1843, Edinburgh; the Scottish mathematician and astronomer who invented the eidograph, a more complicated version of the pantograph used to make scale images of drawings. He was a protegee of John Playfair, and teacher to Mary Somerville. He wrote about the line in 1799. He is also not credited for his 1807 proof of a result about polygons with an equal area, which has become the Bolyai–Gerwien theorem. He was also one of the first in England/Scotland to promote the calculus as taught on the Continent.  


The theorem says that if a triangle is inscribed in a circle, then if perpendiculars are dropped from a point on this circumcircle to the three sides of the triangle (extended as needed) the feet of these perpendiculars will lie on a straight line. It works the other way too. If you draw a straight line cutting all three sides of the triangle, perpendiculars drawn at these points of intersection will be concurrent at a point on the circumcircle.  (With dynamic geometry software, it is relatively easy for students/teachers to create a single line through three sides of a triangle, then construct perpendiculars to the three intersections and make their intersection a traceable point, the rotate the line about ther middle point of the three to get the circumcircle.)

I mentioned recently in a description of David Well's new book, Games and Mathematics, that I keep finding out new stuff. Well, he pointed out a connection between the Wallace line (he uses Simson, but I believe he knows better) and tangents of a parabola.

If you find three tangents to parabola and construct the circumcircle to the triangle formed by their mutual intersections, the circumcircle will pass through the focus of the parabola.
Tricky and cool, but what does that have to do with the the Wallace line? Well if you drop a perpendicular from the focus to ANY tangent, the foot of the perpendicular will always fall on the line tangent to the parabola at the vertex. The tangent at the vertex is a Wallace line for any triangle formed by three tangents to a parabola.


 

Wednesday, 26 April 2023

Sang-Heronian Triangles and some History about Near Equilateral Triangles

 

The first Near Equilateral triangles with consecutive integer sides and integer area (sometimes called Brahamagupta triangles) was discovered over 2500 years ago. The discovery of the 3,4,5 right triangle seems lost in antiquity back before 500 BC. All Pythagorean triangles are Heronian, but lots (infinitely many) of other triangles that are not right triangles also are Heronian.  The second near equilateral triangle, the 13, 14, 15; was known to Heron of Alexandra as early as 70 AD, almost 2000 years ago. Since then, they've grown in number, and to infinity, and been dissected and diagnosed repeatedly. They've even been generalized to three dimensions in Heronian Tetrahedra. Here is one part of their story.

Heron of Alexandria is known to have developed a method of finding the area of triangles using only the lengths of the three sides. It is known that it was proven in his Metrica around 60 AD. His proof was extended in the 7th century by Brahamagupta extended this property to the sides of inscribable quadrilaterals. Since around 1880, the triangular method of Heron has been known as Heron's formula, or Hero's Formula. It emerged in French, formula d'Heron (1883?) and German, Heronisch formel (1875?) and in George Chrystal's Algebra in 1886 in England. 


L E Dickson's History of Number Theory states that Heron stated the 13, 14, 15 triangle and gave its area as 84, the height of 12 being the common side of a 5,12,13 triangle and a 9, 12, 15.  The 5 and 9 combining to form the base of length 14. Brahmagupta is cited in the same work for giving an oblique triangle composed of two right triangles with a common leg a, stating that the three sides are \( \frac{1}{2}(\frac{a^2}{b}+ b)\) , \( \frac{1}{2}(\frac{a^2}{c}+ c)\), and \( \frac{1}{2}(\frac{a^2}{b}- b) + ( \frac{1}{2}(\frac{a^2}{c}- c)\)

In 1621 Bachet took two Pythagorean right triangles with a common leg, 12, 35, 37 and 12, 16, 20 and produced a triangle with sides of 37, 20, 51. With an area of 306 if I did my numbers right.
Vieta and Frans van Schooten, both used the same approach of clasping two right triangles with a common leg; and by the first half of the 18th century, the Japanese scholar, Matsunago, realized that any two right triangles would work, by simply multiplying the sides of each by the hypotenuse of the other, he could juxtapose the two resulting triangles.

In the early 1800's through 1825 the problem was alive and hopping on the Ladies Diary and the Gentleman's Math Companion. One method created right triangles in another triangle to be reassembled into a rational triangle, similar in fact, to the problem that would appear in the 1916 American Mathematical Monthly. (Note; any triangle with rational sides and area can be scaled to become a Heronian triangle.)

In a letter of Oct 21, 1847; Gauss to H. C. Schumacher, he stated a method using circumscribed circles, and found lots of others chose the exact same solutions in their response. E. W. Grebe tabulated a set of 46 rational triangles in 1856. W. A. Whitworth noticed that the 13, 14, 15 triangle of antiquity, that had an altitude of 12, was the only one in which the altitude and sides were all consecutive. (1880)

Somehow, among all those, the contributions of a Professor from Scotland was not observed by Dickson.

The first modern western article I can find on the topic of Near Equilateral triangles with integer sides and area is from Edward Sang which appeared in 1864 in the Transactions of the Royal Society of Edinburgh, Volume 23. I find it interesting that this is only a small aside in a much larger article and that he begins with an approach to examining the angles. Then he arrives at the use of a Pell type equation for approximating the square root of three, \(a^2 = 3x^2 + 1 \) and shows that every other convergent in the chain of approximations is a base of a Near Equilateral Triangle, using sides of consecutive integers. The alternate convergents we seek are given by 2/1, 7/4, 26/15, 97/56... each approaching the square root of three more closely, but also each with a numerator that is 1/2 the base of triangle with consecutive integers for sides and integer area. Perhaps it is easier to just use the recurent relation, \(n_i=4n_{i-1} - n_{i-2}\) with \(n_0=2\), and \(n_1=4\) for the actual middle side,2, 4, 14, 52, 194.... The first few such triangles have their even integer base as2x1=2; (1, 2, 3) area 0; 2x2=4; (3, 4, 5); area 12; 2x7=14;  (13, 14, 15); area 84; 2x26=52; (51, 52, 53); area 1170... etc. Throughout, he refers to "trigons" rather than triangles, and never invokes the name of Heron throughout.  



The next paper using consecutive integer sides was in 1880 by a German mathematician named Reinhold Hoppe, who produced a closed form expression for these almost equilateral Heronian Triangles that was similar to, \( b_n =(2+2\sqrt{3})^n + (2-2\sqrt{3})^n \). His paper calls them "rationales dreieck" (rational triangles) I have not seen the entire paper, and don't know if the term Heronian appeared, or not.

The first American introduction to the phrase "Heronian Triangles", seemed to be an article in the American Mathematical Monthly which posed the introduction as a problem, to divide the triangle whose sides are 52, 56, and 60 into three Heronian Triangles by lines drawn from the vertices to a point within. The problem was posed by Norman Anning, Chillwack, B.C. It then includes a description that suggests it is introducing a new term, "The word Heronian is used in the sense of the German Heronische (with a German citation) to describe a triangle whose sides and area are integral. 

 The only other mentions of a Heronian triangle in English in a google search before the midpoint of the 20th century revealed a 1930 article from the Texas Mathematics Teacher's Bulletin. It credits a 1929 talk, it seems, by Dr. Wm. Fitch Cheney Jr. who, "discusses triangles with rational area K and integral sides a, b, c, the g.c.f of the sides 1, under the name Heronian triangles." (Dr Cheney published an article in the American Mathematical Monthly in 1929, The American Mathematical Monthly, Vol. 36, No. 1 (Jan., 1929), pp. 22-28)  Since any such rational area can be scaled up to an appropriate integer area with integer sides these address the general Heronian Triangle, but still no Near Equilateral, or at least not revealed in the snippet view.  

By the 1980's an article in the Fibonacci Quarterly found a way to produce a Fibonacci like sequence, a second order recursive relation to produce the even bases. Letting \(B_0 = 2, and B_1 = 4\), the recursion was \( U_{n+2} = 4 U{n+1} + U_n\) . This paper by W. H. Gould of West Virginia University addresses the full scope of consecutive sided integer triangles and mentions Hoppe, but not Professor Sang.  Gould's paper seems to be his solution to a problem he had posed earlier in the Fibonacci Quarterly, "of finding all triangles having integral area and consecutive integral sides."  (H. W. Gould, Problem H-37, Fibonacci Quarterly, Vol. 2 (1964), p. 124. .) 
Gould also mentions two other, seemingly earlier posed problems in other journals which I have yet to explore, and given the opportunity, will do so and return to this spot,  If you are impatient, they are

7. T. R. Running, Problem 4047, Amer. Math. Monthly, Vol. 49 (1942), p. 479; Solutions by W. B. Clarke and E. P. Starke, ibid. , Vol. 51 (1944), pp. 102-104.

8. W. B. Clarke, Problem 65, National Math. Mag. , Vol. 9 (1934), p. 63

Gould's article is a wonderful read for the geometry of the incircles and Euler lines in such special triangles is well explored.


These are each candidates to be the first American proposal of these consecutive integer sided triangles, but it seems Gould's paper was the first to expand the full scope of the solutions in any detail.


Some of the characteristics of these I think would be found interesting to HS and MS age students I will spell out below.  

As mentioned above, the length of the middle (even) side follows a 2nd order recursive relation \(B_n = 4B_{n-1}-B_{n-2}\)  so the sequence of these even sides runs 2, 4, 14, 52, 194, 724..... etc. ) is there to represent the degenerate triangle 1,2,3.

Interestingly, the heights follow this same recursive method giving heights of 0, 3, 12, 45, 168....

The height divides the even side into two legs of Pythagorean triangles that make up the whole of the consecutive integer triangle.  They are always divide so that one is four greater than the other, or each is b/2 =+/- 2.

Of the two triangles formed by on each side of the altitude, one is a primitive Pythagorean triangle, PPT, and the other is not.  The one that is a PPT switches from side to side on each new triangle, alternately with the shorter leg, and then the longer leg.  Here are the triangles with the two subdivisions of them with an asterisk Marking the PPT:


Short    Base         Long                      small triangle                 large triangle
  3          4               5                                               *3   4   5
13        14             15                            *5, 12, 13                        9, 12, 15
51        52              53                            24, 45, 51                      * 28, 45, 53
193     194           195                          *95, 168, 193                     99, 168, 195
723     724           725                            360, 627, 723                *364, 627, 725

The pattern of the ending digits of 3, 4, 5 repeated twice, and 1,2,3 once  by looking at the end number behavior of if the previous two numbers end in 4's or a four followed by a two.

In the 1929 article mentioned above, Dr. Cheney writes that he knows of no examples of Heronian triangles up to that time that were not made up of two right triangles, and then gives an example of one that is not decomposable,  25, 34, 39.   He also points out that the altitudes of Heronian triangles are not always integers, and gives the example of 39,58,95 as an example which I calculate to be 4.8.

A paper by Herb Bailey and William Gosnell in Mathematics Magazine, October 2012 demonstrates Heronian triangles in other arithmetic progressions from the near-equilateral ones.

I mentioned that there are also Heronian Tetrahedra, although that use of Heronian seems even later than for triangles, perhaps as late as 2006.   The earliest example of an exact rational tetrahedra with all integer edges, surfaces and volume was by Euler.  He created a tetrahedron formed by three right triangles  parallel to the xyz coordinate axes, and one oblique face connecting them.  The triple right angle edges were 153, 104, and 672, and the three edges of the oblique face were 185, 680, and 697.  These were each Pythagorean right triangles, the four faces of  (104,672,680), (153,680,697), (153,104,185) and (185,672,697)  

There are an infinite number of these Eulerian Birectangular tetrahedra, but they seem to get very large very quickly.  Euler showed that they can be found by deriving the three axis-parallel sides a, b, and c by using four numbers that are the equal sums of two fourth powers.  Euler found an example using , and that's the easy part.  Then he constructed the three monster lengths of 386678175332273368, and 379083360, Yes, those numbers are each in the hundreds of millions, and each pair had a larger hypotenuse to form a third side. 
And as the near end of the Wikipedia discussion of these states, "A complete classification of all Heronian tetrahedra remains unknown."   

Monday, 24 April 2023

Pythagorean Parabolas

 





I recently came across a note on an Annual Meeting of the Rocky Mountain Section of the MAA in 1923. Among the list of presentations was one by W. J. Hazzard, Professor at the Colorado School of Mines on the topic of "Parabolic Grouping of Pythagorean triangles."
I was a little familiar with Prof. Hazard as I had leapfrogged off one of his old posts in the Mathematics Teacher on methods of solving a quadratic equation to write a little about the history of solving quadratics in twenty or so different ways, which I hope someday to reduce to blog posts, but not today. 
I even had a copy of one of the good Professor's books in my collection of old math books, but I had not read, nor was I aware of the idea he spoke of.  With a few words of guidance from a "very" brief coverage in the article, I was able to extract at least a little that may be of interest to anyone who enjoys Pythagorean relations, and especially if you teach high school math.

If you put one acute vertex of a right triangle at the origin and lay it out so that the shorter leg lies along the positive x-axis, the other vertex will be at the point (a,b) as determined by the two legs of the triangle.  In the graph I have shown the  position of a 3-4-5 triangle and a 5-12-13 triangle to make my meaning clear.

A natural question is, "So What?"  But if we look at several of the points determined by the upper vertex, and select out only some "related" Pythagorean triples, we notice a pattern.  In the image at right the points represent the set of triples 3-4-5; 5-12-13; 7-24-25; and 9-40-41.  (Any teacher or student who is not aware, there is a simple trick to find an infinite number of these triangles with a longer side one less than the hypotenuse.  Just take any odd number to be the short leg, square it, and then divide by two and round up to the next whole for the hypotenuse. For example, 11 is a good odd number, and its square is 121.  If we divide 121 by 2 we get 60.5, which is between 60 and 61, and 11, 60, 61 is a Pythagorean triple.) 
All the points lie on a parabola y= 1/2 x2 - 1/2 .  Since the focal length is 1/(4A), with A = 1/2, the focus must be a distance of 1/2 unit above the vertex, making the focal point at the origin. If we think about the definition of a parabola as the set of points equally distant from a focus and directrix, we realize the line of the directrix must be the line y = -1 so that, for instance the point (3,4) which is 5 units from the origin/focus will also be 5 units away from the directrix.  

Admittedly that is a pretty small (although infinity large) sub-set of the Pythagorean triples.  What would happen if we plotted other triangles like 8-15-17?  It turns out they are not on the parabola drawn... they are on another one.  In fact, all the triangles which have a longer leg two less than the hypotenuse will also have a focus at the origin, and the directrix will be ... yeah you knew it would be, y = - 2.  That makes the focus at (0,-1).   You can write the equation with ease for the parabola passing through any of these Pythagorean vertices, and all the ones with a common difference between the longer leg and hypotenuse share a parabola.

All the triples I've picked so far have been primitive triples.  A good question to ask is what would happen if we picked, say, a 6-8-10 triangle. Will it fall on the same parabola as the 3-4-5, or on the ones with a difference of two?

The image below shows parabolas for differences of 1, 2, and 8 between the longer side and hypotenuse, and point D is the 6-8-10 triangle, right there with 8-15-17 and others like it. 





I'm not sure you can swap this information for bread or ale at the local inn, but it's pretty interesting stuff.

Saturday, 22 April 2023

Heron's Cube Root Method


 

As is my nature, I love to roam through old math journals, and recently I found a 1920 article on a beautiful hand calculation for cube roots by Heron from his long lost Metrica.  Until nearly the end of the Nineteenth Century, Math Historians were writing that the ancient Greeks could not calculate square roots (they seldom even mentioned the thought of a cube root) because, "We can't find the evidence so they didn't know it." The discovery of Heron's Metrica in 1897 provided the evidence with several examples of square roots, and one single example of a Cube root.



This is a pretty accurate cube root for 2000 years ago, accurate to the third decimal place, and an error in the root of a little less than 0.0013  that's the total error. If you read it carefully, a question that has plagued all who have studied it.  What is the origin of the 100 added to 180 to get 280?  Some early researchers thought it was the original number, 100, and dismissed Heron's method as not very accurate for many numbers.  But this article provides an explanation that is, in the words of the author, closer than seven digit logarithms for large numbers (he didn't define).  

After doing a few by hand, I "cheated" and built a spreadsheet to compute them and give the error.  But if you try a few by hand, you will agree that it is way ahead of that thing they call the cube root algorithm in YouTube videos.  So I'll let You in on the secret, (but don't tell the other kids!).

Heron's method begins with the surrounding roots of the number.  For the 100 example he used, the lower root bound is 4, and the upper is 5, that is, the number 100 is bound by 64 and 125.   Now we need to determine what Heron called the excess, and the deficit.  The excess is 5^3 - 100 or 25.  The deficit is 100 - 4^3 = 36.  


Then we form the numerator of the fractional part of the solution, multiply the deficit by the upper root bound, 5, producing 180.  

Now for the denominator, we take the 180 from the numerator, and add the product of the excess times the lower bound, producing the unexplained 100.  To get our final root estimate, we add the lower root, 4 to this fraction.  My calculator gives 4.642857 but for the actual cube root of 100 it gives  4.6415888..   That's a total error of a smidge more than 00126.  If you cube the approximation you get 100.08199.   

I ran a spreadsheet for numbers from 10 to 330 and notice a few quirks I believe.  The error seems greater when the deficit is more than the excess.  Which actually makes 100 look worse than average.  If you try 90 instead, where the excess and deficit are 35 and 26, almost the opposite of 100 you get much greater accuracy.   For 90, the true value is 4.481404747, and the estimate is 4.481481481, and the total error is 0.0000076734.  The cube of the estimate is 90.0046  (What we country folk call Pretty Dang Good!)

It also seems that as the numbers get bigger, the errors get slightly smaller, and I can believe the 7 digit logarithm quote.  

So if you are driving down the road and factoring the license plate number in front of you is trivial (Oh, Come On, you know you all do that!)  try taking the cube root in you head.  Fun on the Highway .... (I missed what turn off??????)

ADDEDUM, With a Hat Tip to Keith Raskin;

So if you only read really old journals, you may miss something really important.  Case in point, the 2017 Bulletin of Parnas Mathematical Society, which is in Brazil (Brasil)   It

The Bulletin has an extension of Heron's root taking method for any odd root.   I've only just begun playing with it, but I'll add the method here, and my results for 5th roots.  

The system is much the same, but with a power used in the multiplication of the roots and the excess and deficiency, and the first new task is to find out the power k we want to use in those products.  The key is to let the root you seek, n, be equal to 2k+1; so for the fifth root, since 2k+1 = 5 gives us k=2, we want to square the lower and upper bounds of the root when we multiply.  

Example   Find the fifth root of 100 (which your calculator will tell you is 2.51188...  ) .   

The bounding roots are 2 and 3.  The deficit is 100 - 2⁵ = 68, and the excess is 3⁵- 100 = 143.  Now we proceed as before, except the numerator will be 3² times the deficit, 3² x 68 = 612.

 Now for the denominator we find the product we add to the numerator, which is the square of the lower root bound, 2, times the excess, 143, 2² x 143 = 572.  So the fractional part of our estimate will be 612 / (612 + 572) ... which is 153/296, or as a decimal, 0.51689.  Adding our lower root bound we get 2.051689. An error of 0.00500.  




Tuesday, 18 April 2023

Pandigital Primes

  

No reason for Wells' book on the cover, I just like the picture.  (and his writing)

In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. In base ten such a number might be 123456789098765444321.  If the number is prime, which is really cool, it is called a pandigital prime.  And if it uses the digits exactly once each, which is even cooler, .... Unfortunately, in base ten,  which is where a lot of us hang out the most, you can't have such a number.  Any ordering of 1,2,3,4,5,6,7,8,9,and 0 will be divisible by three, and hence - NOT prime.  Even if you leave out the zero, you can't make one with the first nine digits either for the same reason.( I know..."Ahhhhh".)(But you can have all of 1 through 9 if the tenth digit is not zero but 1, ....  1234567891 is Prime.)

So there are a couple of ways to adjust.  We can look for primes that are n digits long and use the first n numerals, for example 2143 is a four digit prime using the numerals 1,2,3, and 4.  The problem with this approach is that there are only two of them.  The four digit one is shown, and a seven digit one is 7652413 . Any number made up of the first  2, 3, 5, 6, 8, 9, or ten digits will be divisible by three.

That leaves a couple of options.  I got started thinking about these when I wrote a blog awhile back called "The Game of Primes ."  The object was to create a string of primes by starting with one prime number and then adding a digit each time to make the string a longer prime, but using any of the ten decimal numerals as a digit.  So you could start with 2, then add 3 to get 23, etc.  I only got to seven, you may be able to do better.  There is a nine digit prime (several of them) that has no repeated numerals.  I found 576849103 is prime  and  so is 987654103.  Having pretty much reached the ends of my manual calculating limits, I asked on twitter, "Is there a nine digit prime using distinct digits that includes a two?"
Faster than a nano-bullet I got a response from jomo ‏ @n0m0 who advised me that "First nine digit prime with distinct digits that includes number 2 is 102345689, second is 102345697 and so on again!"  

Realizing I had a computation wizard on the line (at least relative to me) I wondered aloud, (or Atweet) 
"Is it possible to form an eleven digit Pandigital Prime (ie repeating only one of the 0-9)"  Again at something akin to the speed of light,  he responded with two examples; "First pandigital prime with 11 digits is 10123457689, next one is 10123465789 and so on..." 

Then, realizing he had a rube on line whose non-programming nose could easily be pushed in the mud, he sent me a list of several... you can count them, and lock this away if you are looking for 11-digit primes... 

Here is the first few, but the whole list he has graciously placed here.  List of 11 digit pandigital primes filtered from ~10 million primes

10123457689
10123465789
10123465897
10123485679
10123485769
10123496857
10123547869
10123548679
10123568947
10123578649
10123586947
10123598467
10123654789
10123684759
10123685749
10123694857
10123746859
10123784569
10123846597
10123849657
10123854679
10123876549
10123945687
10123956487
10123965847
10123984657
10124356789
10124358697
10124365879
10124365987

Thursday, 13 April 2023

On This Day in Math - April 13

  

Double False Position from Gemma Frisius  Arithmeticae Practicae Methodus Facilis (1540) *MAA 

"I endeavor to keep their attention fixed on the main objects of all science, the freedom & happiness of man, so that coming to bear a share in the councils and government of their country, they will keep ever in view the sole objects of all legitimate government."
A plaque with this quotation, with the first phrase omitted, is in the stairwell of the pedestal of the Statue of Liberty.
~Thomas Jefferson, in  a Letter to Tadeusz Kosciuszko, 26 February 1810

The 103rd day of the year; there are 103 geometrical forms of magic knight's tour of the chessboard.

103 is the reverse of 301. The same is true of their squares: 1032 = 10609 and 3012 = 90601. *Jim Wilder

The smallest prime whose reciprocal contains a period that is exactly 1/3 of the maximum length. (The period of the reciprocal of a prime p is always a divisor of p-1, so for 103 the period is 102/3 = 34. )

Using a standard dartboard, 103 is the smallest possible prime that cannot be scored with two darts.

Most mathematicians know the story of 1729, the taxicab number which Ramanujan recognized as a cube that was one more than the sum of two cubes, or the smallest number that could be expressed as the sum of two cubes in two different ways.  But not many know that 103 is part of the second such pair  \(64^3 + 94^3 = 103^3 + 1^3  \)

*************** Lots of additional math facts for days 91-120 at https://mathdaypballew.blogspot.com/


EVENTS

1560  On this day in 1560, Cardan's son Giambatista was executed after being found guilty of poisoning his wife. This was a blow from which Cardan never recovered.
==============================================================
In 1620, the word "microscope" was coined as a suggested term in a letter written by Johannes Faber of Bamberg, Germany, to Federigo Cesi, Duke of Aquasparata and founder of Italy's Accademia dei Lincei (Academy of the Lynx). This Academy, possibly the world's first scientific society took its name after the animal for its exceptional vision. *TIS   (Galileo had called it the occhiolino 'little eye').

1668 Lord Brouncker, President of the Royal Society, publishes "the fist mathematical result to be published in a mathematical journal" in the Philosophical transactions of the Royal Society. His demonstration of the method of quadrature of the rectangular hyperbola, y= x-1 extended the work of Wallis in Arithmetica infinitorium. Brouncker had been working with Wallis on extending the work of Torricelli's Opera geometrica hoping to apply the methods to the long-sought quadrature of the circle.
The rectangular hyperbola had eluded Fermat, and only been partially solved by de Saint Vincent by 1625. It was a fellow Jesuit of Saint Vincent, Alphonse Antonio de Sarasa he may have been the first to recognize that certain areas under the hyperbola are related to each other in the same was as logarithms. *Jacqueline Stedall, Mathematics Emerging, 2008.


1672 After presenting his paper on the composition of light as a, “heterogeneous mixture of differently refrangible rays” on 19 Feb, several critics emerged, most notably Robert Hooke. Newton responded to the critiques with a letter to the Royal Society, "Some Experiments propos'd in relation to Mr. Newtons Theory of light, printed in Numb. 80; together with the Observations made thereupon by the Author of that Theory; communicated in a Letter of his from Cambridge, April 13. 1672." Newton had performed a series of experiments to validate his theory, and here described the results. See the letter here.

Halley's Comet, March 8, 1986
1759  Halley’s comet returns, as he predicted in 1682. The comet last reached perihelion on 9 February 1986, and will next reach it again on 28 July 2061 *Wik   Halley's prediction that it would return in 1758 was incorrect, and observations and calculations led to a correct prediction and perihelion occurred on April 13, 1759.  It was sighted in 1758, the year he predicted, on 25 December, when it was observed by German farmer, and armature astronomer, Johan Palitsch. *HT to @RMathematicus


1791 Legendre is named one of the French Academy’s three commissioners for the astronomical operations and triangulations necessary for determining the standard meter. The others were Mechain and Cassini IV. [DSB 8, 136]*VFR


BIRTHS

953 Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji (13 April 953 in Baghdad (now in Iraq) - died about 1029) Al-Karaji was an Islamic mathematician who wrote about the work of earlier mathematicians and who can be regarded as the first person to free algebra from geometrical operations and replace them with the type of operations which are at the core of algebra today. *SAU

1728 Paolo Frisi (13 Apr 1728; 22 Nov 1784 at age 56) Italian mathematician, astronomer, and physicist who is best known for his work in hydraulics (he designed a canal between Milan and Pavia). He was, however, the first to introduce the lightning conductor into Italy. His most significant contributions to science, however, were in the compilation, interpretation, and dissemination of the work of other scientists, such as Galileo Galilei and Sir Isaac Newton. His work on astronomy was based on Newton's theory of gravitation, studying the motion of the earth (De moto diurno terrae). He also studied the physical causes for the shape and the size of the earth using the theory of gravity (Disquisitio mathematica, 1751) and tackled the difficult problem of the motion of the moon. *TIS

1743 Thomas Jefferson, American President and Mathematical enthusiast, was born.
"Thomas Jefferson had four main accomplishments in mathematics. First, he took mathematics from the ranks of a secondary subject and raised it to such a prominence in the curriculum of the University of Virginia that it was not seen at any other college in the United States at the time. Through Jefferson’s influence, men like J.J. Sylvester, in 1841 (though unsuccessful), were recruited to build up the mathematics courses at the University of Virginia....David Eugene Smith sums it best in the following passage:
It is apparent that Jefferson was not a mathematician but that he was a man who appreciated the beauties, the grandeur, the values, the classics, and the uses of mathematics and did much to give to the science a recognized standing as a university subject. "
From an online article by Ajaz Siddiqui, See the short article here.

1802 George Palmer Williams (Woodstock, Vermont, April 13, 1802-Ann Arbor, September 4, 1881) Hegraduated Bachelor of Arts from the University of Vermont in 1825, and then studied about two years in the Theological Seminary at Andover, Massachusetts. He did not complete the course, but took up teaching, which proved to be his life work.
He was Principal of the Preparatory School at Kenyon College, Ohio, from 1827 to 1831. In 1831 he was elected to the chair of Ancient Languages in the Western University of Pennsylvania, but after two years he returned to Kenyon College, where he remained until called, in 1837, to the branch of the incipient University of Michigan at Pontiac.
In 1841, when the College proper was opened at Ann Arbor, he was made Professor of Natural Philosophy. In 1854 he was transferred to the chair of Mathematics and in 1863 to the chair of Physics. From 1875 to 1881 he was Emeritus Professor of Physics.
He received the degree of Doctor of Laws from Kenyon College in 1849. The University Senate in a memorandum relative to his death declared that: "Dr. Williams welcomed the first student that came to Ann Arbor for instruction; as President of the Faculty he gave diplomas to the first class that graduated, and from the day of his appointment to the hour of his death his official connection with the University was never broken."
In 1846 he was ordained to the ministry of the Protestant Episcopal Church; but he did no regular parish work, except for a short time in Ann Arbor. He was first and last a teacher, beloved by his colleagues and pupils and universally respected and honored.
Some years before his death the alumni raised a considerable fund, the proceeds of which were to be paid to him during his lifetime and after his death were to be used for maintaining a professorship named in honor of his memory. *Hinsdale and Demmon, History of the University of Michigan 221

1813 Duncan Farquharson Gregory (13 April 1813 in Edinburgh, Scotland - 23 Feb 1844 in Edinburgh, Scotland) Scottish mathematician who was one of the first to investigate modern ideas of abstract algebra.In this work Gregory built on the foundations of Peacock but went far further towards modern algebra. Gregory, in his turn, had a major influence on Boole and it was through his influence that Boole set out on his innovative research. *SAU

1869 Ada Isabel Maddison (April 12, 1869 - October 22, 1950) born in Cumberland, England. She attended Girton College, Cambridge, in the same class with Grace Chisholm Young and they attended lectures of Cayley. Then she came to Bryn Mawr, where she earned her Ph.D. in 1895. She continued there until retirement, involved mostly in administrative work. *WM

1879 Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematicianborn in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry. Together with Federigo Enriques, he won the Bordin prize from the French Academy of Sciences.
He contributed in a major way to birational geometry, the theory of algebraic surfaces, in particular of the curves lying on them, the theory of moduli spaces and the theory of functions of several complex variables. He wrote prolifically, and some of his work has subsequently been shown to be not rigorous according to the then new standards set in particular by Oscar Zariski and David Mumford. At the personal level, according to Roth (1963) he was easily offended, and he was involved in a number of controversies. He died in Rome of cancer.*Wik

1889 Herbert Osborne Yardley American cryptographer who organized and directed the U.S. government's first formal code-breaking efforts during and after World War I. He began his career as a code clerk in the State Department. During WW I, he served as a cryptologic officer with the American Expeditionary Forces in France during WWI. In the 1920s, when he was chief of MI-8, the first U.S. peacetime cryptanalytic organization, he and a team of cryptanalysts exploited nearly two dozen foreign diplomatic cipher systems. MI-8 was disbanded in 1929 when the State Department withdrew funding. Jobless, Yardley caused a sensation in 1931 by publishing his memoirs of MI-8, The American Black Chamber, which caused new security laws to be enacted.*TIS


1905 Bruno Rossi (13 Apr 1905, 21 Nov 1993)Italian pioneer in the study of cosmic radiation. In the 1930s, his experimental investigations of cosmic rays and their interactions with matter laid the foundation for high energy particle physics. Cosmic rays are atomic particles that enter earth's atmosphere from outer space at speeds approaching that of light, bombarding atmospheric atoms to produce mesons as well as secondary particles possessing some of the original energy. He was one of the first to use rockets to study cosmic rays above the Earth's atmosphere. Finding X-rays from space he became the grandfather of high energy astrophysics, being largely responsible for starting X-ray astronomy, as well as the study of interplanetary plasma. *TIS

1909 Stanislaw M. Ulam (13 Apr 1909; 13 May 1984 at age 75) Polish-American mathematician who played a major role in the development of the hydrogen bomb at Los Alamos. He solved the problem of how to initiate fusion in the hydrogen bomb by suggesting that compression was essential to explosion and that shock waves from a fission bomb could produce the compression needed. He further suggested that careful design could focus mechanical shock waves in such a way that they would promote rapid burning of the fusion fuel. Ulam, with J.C. Everett, also proposed the "Orion" plan for nuclear propulsion of space vehicles. While Ulam was at Los Alamos, he developed "Monte-Carlo method" which searched for solutions to mathematical problems using a statistical sampling method with random numbers. *TIS He is buried in Santa Fe National Cemetery in Santa Fe, New Mexico, USA

DEATHS

1728 Samuel Molyneux (18 Jul 1689, 13 Apr 1728 at age 38)British astronomer (Royal Observatory at Kew) and politician. Together with assistant James Bradley, he made measurements of abberation - the diversion of light from stars. They made observations of the star  Draconis with a vertical telescope. Starting in 1725 they had the proof of the movement of the earth giving support to the Copernican model of the earth revolving around the sun. The star oscillated with an excursion of 39 arcsecs between its lowest declination in May and its the highest point of its oscillation in September. He was unfortunate to fall ill in 1728 and into the care of the Anatomist to the Royal Family, Dr Nathaniel St Andre, whose qualifications were as a dancing master. Molyneux died shortly thereafter. *TIS

1906 Walter Frank Raphael Weldon DSc FRS (Highgate, London, 15 March 1860 – Oxford, 13 April 1906) generally called Raphael Weldon, was an English evolutionary biologist and a founder of biometry. He was the joint founding editor of Biometrika, with Francis Galton and Karl Pearson.*Wik Pearson said of him, "He was by nature a poet, and these give the best to science, for they give ideas." *SAU

1941 Annie Jump Cannon (11 Dec 1863; 13 Apr 1941) American, deaf astronomer who specialized in the classification of stellar spectra. In 1896 she was hired at the Harvard College Observatory, remaining there for her entire career. The Harvard spectral classification system had been first developed by Edward C. Pickering, Director of the Observatory, around the turn of the century using objective prism spectra taken on improved photographic plates. In conjunction with Pickering Cannon was to further develop, refine, and implement the Harvard system. She reorganized the classification of stars in terms of surface temperature in spectral classes O, B, A, F, G, K, M, and cataloged over 225,000 stars for the monumental Henry Draper Catalog of stellar spectra, (1918-24).*TIS

2004 David Herbert Fowler (April 28, 1937 – April 13, 2004) was a historian of Greek mathematics who published work on pre-Eudoxian ratio theory (using the process he called anthyphaeresis). He disputed the standard story of Greek mathematical discovery, in which the discovery of the phenomenon of incommensurability came as a shock.
His thesis was that, not having the real numbers, nor division, the Greeks faced difficulties in defining rigorously the notion of ratio. They called ratio 'logos'. Euclid Book V is an exposition of Eudoxus's theory of proportion, which Eudoxus discovered about 350BC, and which has been described as the jewel in the crown of Greek mathematics. Eudoxus showed by a form of abstract algebra how to handle rigorously the case when two ratios are equal, without actually having to define them. His theory was so successful that, in effect, it killed off perfectly good earlier theories of ratio, and Fowler's aim had been to find the evidence for the rediscovery of these previous theories.
In particular Thaetetus (c 414-369BC) introduced a definition of ratio using a procedure called anthyphairesis, based on the Euclidean subtraction algorithm. Fowler developed his ideas in a series of papers, culminating in the book The Mathematics of Plato's Academy: A New Reconstruction, which was published in 1987. This book is based on a study of the primary sources and on their assimilation and transformation.*Wik

2008 John Archibald Wheeler (9 Jul 1911, 13 Apr 2008 at age 96) was the first American physicist involved in the theoretical development of the atomic bomb. He also originated a novel approach to the unified field theory. Wheeler was awarded the 1997 Wolf Prize "for his seminal contributions to black hole physics, to quantum gravity, and to the theories of nuclear scattering and nuclear fission." After recognizing that any large


Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell