**Henry Heaton**'s parents were Weaver Heaton (1810-1873) and Rebecca Naylor Sharp (1816-1879). Weaver Heaton, the son of Jacob Bowen Heaton (1786-1839) and Rachel Weaver (1787-1858), married Rebecca, the daughter of Isaac Sharp and Eliza Ann Naylor, on 26 October 1834 in Pennsylvania. Weaver was a millwright who designed and built mills, and made mill machinery. Weaver and Rebecca Heaton had eight children: Emaline Heaton (born 1835); James Heaton (born 1838); Rachel Heaton (born 1840); Eliza Heaton (born 1842); Henry C Heaton (born 1846, the subject of this biography); William Naylor Heaton (born 1847); George Heaton (born 1851); and Ella M Heaton (born 1851). We note that George Heaton and Ella M Heaton were twins. At the time of the 1850 census (taken on 9 August of that year), John H Sharp aged 19 was also living with the family. Although we are unsure how he was related to the family, it is highly probable (given his surname Sharp) that he was a relation of Henry's mother.

In 1852 the family moved to Greenfield also in Washington County, Pennsylvania. In Greenfield, Henry attended the local [1]:-

The 1860 census was taken on 17 July of that year and at that time Weaver and Rebecca Heaton together with seven of their eight children were still living at their home in Greenfield. James, who was 21 years old at this time was working as a ship's carpenter. Rachel was 18, Eliza 17, Henry 14, William 11, and twins George and Ella were 8 years of age.... school four months every winter until he was fourteen years old. During the remaining portion of the year he attended, sometimes a subscription school, but a good portion of the time the California Seminary, an institution located in the adjoining village of California, and patronized largely by young folks, male and female, who either were teachers or expected to become such. Up to his fifteenth year his educational advantages were very good for the time.

By this time Henry had learn quite a lot of elementary mathematics [1]:-

Henry Heaton began his two parallel careers at the age of eighteen. One of these careers was as a teacher while the other was as a carpenter. With these two jobs he was able to make enough money to allow him to attend Mount Union College in Ohio in the winter of 1866-67 and he studied there for his B.S. which he was awarded 17 June 1869. In September 1869 he moved to Iowa to live with his aunt and uncle John and Ann Sharp in the town of Taylor, Appanoose County, Iowa. At the time of the 1870 census, taken on 29 July of that year, Heaton was living with the Sharps and their eight children aged between 8 and 26 years, who were all living in the family home. Heaton's occupation, as shown on the census form, was that of carpenter. By this time Heaton's parents had left Greenfield and were living in New Haven, Fayette County, Pennsylvania.He commenced the study of Arithmetic at ten, Elementary Algebra at eleven or twelve, Trigonometry in his fourteenth year, and Higher Algebra and the first four books of Geometry were completed just after he was fourteen years old. From the effects of one example, in arithmetic, which his teacher had solved for him, his mathematical talent became aroused, and he was able to solve, unaided, any problem found in the textbooks.

In the autumn of 1864 Joel E Hendricks and his family moved to Des Moines, Polk County, Iowa. There Hendricks became involved in surveying and until 1873 he undertook such work. However, Hendricks had spent his whole life with a great love for mathematics and in 1873 he founded a mathematics journal which he named *The Analyst*. Heaton moved to Des Moines to take up a position as a teacher and there he made the acquaintance of Hendricks who encouraged Heaton's interest in mathematics. Hendricks suggested that Heaton subscribe to the *Yates County Chronicle* and it was to this journal that he began to send solutions of mathematical problems posed in the *Chronicle*. It was Hendricks who taught Heaton the basic concepts of probability theory which he quickly mastered.

The 1st part of the 1st volume of *The Analyst* appeared in January 1874. In the third part of the 1st volume Heaton's solution to the following problem was published:

In Volume 3, Part 3, Hendricks published a solution to the "Four Bricks Problem" and notes that Heaton has solved this problem. This famous problem is as follows:A ball rolls down the convex surface of a fixed sphere, the friction being just sufficient to prevent sliding; find the point where it leaves the sphere.

The problem had first been posed by Artemas Martin in the December 1871 part of theIf four bricks are placed on each other at random, with their longest axes horizontal and in the same vertical plane, determine the probability that the pile will stand.

*Schoolday Visitor*but had not been solved. G H Harvill writes in [1] that Heaton contributed:-

Heaton's contributions to... many elegant solutions of difficult problems in 'The Analyst', 'Mathematical Visitor', 'Mathematical Magazine', 'Mathematical Messenger', 'Annals of Mathematics' and 'School Visitor', to all of which he has been a regular and leading contributor. His many elegant solutions of the most difficult Probability problems places him abreast of the renowned E B Seitz and today he has no superiors and few equals, in this, his favourite field.

*The Analyst*are given at THIS LINK.

After *The Analyst* ended publication in 1883, its continuation was the *Annals of Mathematics*, the 1st part of the 1st volume of which was published in March 1884. Heaton continued his contributions to the *Annals*. For example in 1887 he solved the following problem in the *Annals* posed by Artemas Martin:-

Between 1878 and 1894 Heaton posed, or published solutions to, several problems inEach of four equal circles touches two of the others. Find the average area included between them.

*The Mathematical Visitor*. For example he posed:-

He solved various problems including:-If n people meet by chance, what is the probability that they all have the same birthday, supposing every fourth year a leap year.

He gained fame (and a prize) by solving Artemas Martin's 'Three Pennies' Prize Problem:The first of two casks containing a gallons of wine, and the second b gallons of water. Part of the water is poured into the first cask, and then part of the mixture is poured back into the second. Required the probability that not more then1/n of the contents of the second cask is wine.

Another problem he solved was the famous "Duck Problem":-Three equal coins are dropped horizontally at random, one at a time, into a circular box whose diameter is twice that of one of the coins. Find the chance that only one coin rests on the bottom of the box.

The article [1] was published in 1895 and at this time theA sportsman saw a duck in a circular pond, which dodged behind a tree growing in it. The sportsman ran along the edge of the pond, but the duck kept behind the tree all the time and swam towards the shore. Required the equation of the curve the duck described, and the distance it swam to reach the shore.

*American Mathematical Monthly*had only just begun publication. Heaton posed the following problem in the 1st part of the 1st Volume of the

*Monthly*in 1894, namely:

We list a collection of problems posed by Heaton, and a collection of those solved by Heaton, all of which were published either inThrough two given points to pass four spherical surfaces tangent to two given spheres.

*The Analyst*or in the

*American Mathematical Monthly*, at THIS LINK.

We noted that Heaton met Joel Hendricks when he (Heaton) was teaching at Des Moines in the early 1870s. On 30 March 1875 he married Mary Ann Marker, a daughter of George Marker (1818-c.1877) and Rachel Bitner (1818-1854). Henry and Mary Ann Heaton had two children; Roy Henry Heaton (born 26 June 1876) and Emma Margaret Heaton (born 27 March 1884). Emma died at the age of eight in November 1892.

The Heatons did not stay in Des Moines for long for in 1877-78 they were living in Sabula, Jackson County, Iowa, a town on the Iowa/Illinois border almost surrounded by the Mississippi River, where Heaton was Superintendent of Schools. While there he published the paper [3] on cubic equations. In this paper Heaton solved the cubic equation *x*^{3} - 3*ax* = 2*b* by putting *x* = 2√*a* cos *t*. Then the original equation becomes 2cos 3*t* = 2*b*/(*a*√*a*) so

*x*= 2√

*a*cos{cos

^{-1}[

*b*/(3

*a*√

*a*)]}.

*American Mathematical Monthly*began publishing in 1894 and by this time he had been awarded a Master's Degree.

In 1896 Heaton, who was now County Surveyor, published [2], namely *A Method of Solving Quadratic Equations* in the *American Mathematical Monthly.* In this paper he presented the following method of deriving the "usual" formula for the solution of the general quadratic:

*ax*^{2} + *bx* + *c* = 0

Transposing the middle term we have

*ax*^{2} + *c* = -*bx*

Squaring, *a*^{2}*x*^{4} + 2*acx*^{2} + *c*^{2} = *b*^{2}*x*^{2}

Subtracting 4*acx*^{2},

*a*^{2}*x*^{4} - 2*acx*^{2} + *c*^{2} = (*b*^{2} - 4*ac*)*x*^{2}

Extracting the square root,

*ax*^{2} - *c* = ±√(*b*^{2} - 4*ac*)*x*

Adding equation (2),

2*ax*^{2} = [-*b* ± (√(*b*^{2} - 4*ac*)]*x*

Whence *x* = [-*b* ± (√(*b*^{2}- 4*ac*)]/2*a*.

Heaton asks, "Is this new?"

*Infinity, the Infinitesimal, and Zero*. He begins the paper by writing:-

Later in the article, showing remarkable confidence in his own understanding, he writes:-So much has been written upon this subject that I do not flatter myself that I can write anything new. I shall only attempt to point out a few of the things that have been written that to my mind plainly cannot be true, and with these errors in view endeavour to express the truth as I see it. If in so doing I fall into error I shall have the satisfaction of knowing that greater men have done the same. In the Analyst, Vol. VIII., pages105-113, Professor Judson has a very interesting article upon this subject in which he quotes freely from many distinguished authors many things that are plainly fallacious. Yet when he attempts to outline his view of the subject it seems to the writer that he blunders fully as badly as those whom he criticises.

In 1906 Heaton moved from Atlantic to Belfield, North Dakota. It is unclear exactly how long he lived in Belfield but by 1917 he was back in Atlantic, Iowa. He died in Biddle, Power River County, Montana, at the age of 80 in January 1927. His wife outlived him by over a year, dying in Biddle in June 1928 at the age of 84.I believe that mathematical infinity is something beyond the ken of even a Professor De Morgan, that he was never so far from having a correct conception of it as when he claimed to have a clear conception of it. Tan x does not become infinite as x approaches90°, for so long as x differs by the shadow of a hair from90°, tan x is finite. I think I know exactly what is meant by the term zero. But I can have no conception either of infinity or of the infinitesimal, and I think it would be well if mathematicians would let both pretty severely alone.

**Article by:** *J J O'Connor* and *E F Robertson*

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