**Woolsey Johnson**'s parents were Charles Frederick Johnson (1804-1882) and Sarah Dwight Woolsey (1805-1870). Charles Johnson was a gentleman farmer in Tioga County, the son of Robert Charles Johnson (1766-1806) and Catherine Anne Bayard (1770-1806). Sarah Woolsey was the daughter of William Walton Woolsey (1766-1839), a New York merchant who imported sugar, cotton and hardware, and Elizabeth Dwight (1772-1813). Sarah had an older brother, Theodore Dwight Woolsey (1801-1889), who was the President of Yale College from 1846 to 1871. Charles and Sarah Johnson were married on 23 April 1835. Charles Johnson had graduated from Union College in 1823 and then attended the Litchfield Law School. However, he inherited much land from his father and, a couple of years after his marriage, the family settled on his 200-acre farm Meadow Bank on the west bank of the Owego Creek. Charles and Sarah Johnson had six children: Charles Frederick Johnson was born on 8 May 1836, graduated B.A. from Yale in 1855, became a Civil Engineer and was appointed Professor of Mathematics at the Naval Academy at Annapolis in 1866; Susan Johnson was born on 28 December 1838 but only lived for five months; Anna Muirson Johnson was born in 1840 and married William Bellamy from Boston, Massachusetts, in 1876; William Woolsey Johnson, the subject of this biography, was born in 1841; Catherine Livingston Bayard Johnson was born in 1845; and Nicholas Bayard Johnson was born in 1849 (died young).

The Johnson family were well-off and they had their son Woolsey (as he liked to be known) tutored privately to prepare him for College. He entered Yale College in New Haven, Connecticut in 1858. This College was renamed Yale University in 1864, two years after Johnson graduated with a B.A. He was awarded first prize in mathematics in his first year of study and continued to excel winning the prize for mathematical solutions in his second year. He continued to win prizes in his Junior and Senior years and graduated with a B.A. in 1862. During his four years at Yale College, Johnson wrote letters to his parents, his sister Katherine, and his brother Charles. Many of these letters survive and are held in the Manuscripts and Archives of the Sterling Memorial Library of Yale University.

After Johnson graduated, in 1862 he was appointed as an assistant in the Nautical Almanac office in Cambridge, Massachusetts. He held this position for two years before being appointed as an assistant professor of mathematics at the U.S. Naval Academy at Newport, Rhode Island in 1864. He was only one year at Newport since in 1865 the U.S. Naval Academy moved to Annapolis where he worked until 1870. Johnson married Susannah Leverett Batcheller (1850-1916) on 12 August 1869, in Annapolis. Susannah was the daughter of the Rev. Breed Batcheller and Sarah Miller Leverett, a descendant of John Leverett, Governor of Massachusetts. Woolsey and Susannah Johnson had two children: Charles William Leverett Johnson, born at Gambier, Ohio, 12 August 1870; and Theodore Woolsey Johnson, born at Owego, 4 June 1872. We will say a little more about the two sons below. In the year that Johnson married, 1869, he published the first of a number of textbooks, *An Elementary Treatise on Analytical Geometry, embracing Plane Coordinate Geometry and an Introduction to Geometry of Three Dimensions*. This book is described as:-

See THIS LINK for the Preface to this book.... designed as a textbook for colleges and scientific schools.

In 1870 Johnson moved to Kenyon College in Gambier, Ohio. Kenyon College, the oldest private college in Ohio, was founded in 1824. Johnson and his wife were already living in Gambier when their first child was born in August 1870. He spent two years at Kenyon College before moving to Annapolis in 1872 when he was appointed as a professor of mathematics at St John's College. This College had been founded in 1784 but it had closed during the American Civil War when it was badly damaged. It had reopened in 1866, six years before Johnson took up his position there. It continued to operate a very traditional approach to education with a fixed course which all students followed. Johnson began to collaborate with John Minot Rice, a professor of mathematics in the United States Navy and, in 1874, they produced the First Part of their work *An Elementary Treatise on the Differential Calculus founded on the Method of Rates or Fluxions*. This was:-

The authors write in the Preface:-Printed for the use of the Cadets at the U.S. Naval Academy.

Indeed, they did complete and publish the work in 1879. You can read the Preface of the finished work at THIS LINK.The mode of teaching the Differential Calculus introduced in these pages has been used by the authors for several years past, in the instruction of their classes, with unusually satisfactory results. The treatise, of which this is the first part, will be completed and published as soon as possible; it is designed to embrace all the topics usually found in textbooks on this subject, and to fit the student for the study of treatises in which application is made of the Calculus to physical research.

Clearly with Johnson in Annapolis where he had previously worked with the U.S. Naval Academy, he could continue to keep his links with the Academy despite teaching at St John's College. For example, as we have just seen, he was collaborating with a mathematician in the United States Navy and writing works for Cadets at the Academy. It is perhaps only a natural progression, therefore, that he should move to the U.S. Naval Academy at Annapolis in 1881. He continued in this post until he retired in 1921. However, he did make several trips abroad in the summer vacations and spent a full year sabbatical at the University of Cambridge in England in the academic year 1886-1887. This full year included both the summer of 1886 and the summer of 1887. The whole Johnson family went with him to Cambridge where to took a house for the year. It was during this year at Cambridge that he wrote his book *A Treatise on Ordinary and Partial Differential Equations*. The Preface of this book is given at THIS LINK.

We promised above to give some more details of Johnson's sons. Charles William Leverett Johnson (1870-1954) was awarded a B.A. from Johns Hopkins University in 1891 and then continued to study there for a Ph.D. He was awarded his Ph.D. from Johns Hopkins in 1896 for his thesis *Musical Pitch and the Measurement of Intervals Among the Ancient Greeks*. After Johnson had received a letter of congratulations following his son receiving his doctorate, he replied:-

In fact Charles Johnson became an Instructor in Greek at Yale University publishing his thesis and the bookThanks for your congratulation concerning my boy. I am glad to say he is his mother's boy, but he has mathematics enough in him to have written his thesis on the mathematical theory of music as it appears in the old Greek philosophers, Musical Pitch and the Measurement of Intervals among the Ancient Greeks. A friend critic wrote him a private letter, congratulating him on being at once "a musician, a mathematician, and a Hellenist." - You see I am getting garrulous, if you will accept bragging of his boys as a test of garrulity. - My other boy is more in my own line, as an engineer.

*The motion of the voice: In the theory of ancient music*(1899). Johnson's younger son, Theodore Woolsey Johnson (1872-1953), was as he said in the above quote, more in his own line. He was awarded a B.A. from Johns Hopkins in 1892 after which he went to the Stevens Institute where he received an M.E. in 1896. He became a Steel Inspector for the U.S. Navy in 1898. In 1910 he published, in collaboration with Frank William Bartlett, the book

*Engineering Descriptive Geometry*. The authors begin their Preface as follows:-

Johnson was one of the founders of New York Mathematical Society in 1888. We see something of his character in his article in the October 1891 part of theThe aim of this work is to make Descriptive Geometry an integral part of a course in Mechanical or Engineering Drawing. The older books on Descriptive Geometry are geometrical rather than descriptive. Their authors were interested in the subject as a branch of mathematics, not as a branch of drawing. Technical schools should aim to produce engineers rather than mathematicians, and the subject is here presented with the idea that it may fit naturally in a general course in Mechanical Drawing.

*Bulletin*of that Society where he wrote:-

The New York Mathematical Society became the American Mathematical Society in 1894 and Johnson served on the AMS Council. He was a member London Mathematical Society, a corresponding member of the British Association for the Advancement of Science and a Fellow of American Association for the Advancement of Science.As there is no doubt that our ancestors originated the decimal system by counting on their fingers, we must, in view of the merits of the octonary system, feel profound regret that they should have perversely counted their thumbs, although nature has differentiated them from the fingers sufficiently, she might have thought, to save the race from this error.

Let us say a little more about Johnson's mathematics. We refer the reader to THIS LINK and THIS LINK for Prefaces to Johnson's books. The reader will see from these Prefaces that Johnson's approach was somewhat old-fashioned for the time in which he wrote but, of course, we must remember that basically he was writing to teach mathematics to U.S. Navy cadets. For example, G H Hardy reviewing *A Treatise on the Integral Calculus, founded on the Method of Rates* writes [4]:-

In fairness to Johnson, the book G H Hardy is reviewing is a 1907 version of a book that Johnson did write over 20 years earlier! Here is a review ofIt is no disparagement of Professor Johnson's book to say that it might have been written thirty years ago. It is a book of a frankly old-fashioned type, another Todhunter or Williamson. It may be better or worse than Todhunter; on the whole we prefer it to Todhunter: but it is with such books as Todhunter that we must compare it. We must not be understood as implying that Professor Johnson's book may not be of considerable utility. There is plenty of room for an improved Todhunter.

*A Treatise on Ordinary and Partial Differential Equations*which was published in the year the book was written [1]:-

Cassius Jackson Keyser (1862-1947), reviewing (a late abridged edition of)This treatise on differential equations is in continuation of the series of mathematical textbooks, by the same author, of which have already appeared the differential and integral calculus. Professor Johnson is professor of mathematics at the United States Naval Academy at Annapolis, and it may be that some will trace in the book methods which are said to be characteristic of the United States Army and Navy mathematics; but it must be said that the plan pursued is likely to lead to a clearer understanding by the student. The object is to give a knowledge of the subject, so far as it is likely to have practical application; and in this it is safe to say that Professor Johnson has succeeded.

*An Elementary Treatise on the Differential Calculus founded on the Method of Rates*many years after it was first written, does take a rather "American" approach [5]:-

The anonymous reviewer [2] is full of praise:-Professor Johnson's book is an abridgment of his "Differential Calculus" and is based on the method of rates. It is distinguished by the absence of the histological methods of the rigorists. Things are presented pretty much at their face values, being shown graphically rather than laboriously proved by help of the refined logical machinery brought in from across the sea. The English is excellent, everything moves along with the calmness, dignity and facility of an elder day, and the reader, while acquiring much useful knowledge, will acquire also a degree of confidence that in these critical times is apt to be rare and is apt also to suffer mitigation in the course of subsequent study.

When a mathematician of eminence undertakes thus to provide a treatise upon a subject of importance, and compiles a text-book, for young students, not only the youth who is thus provided with a text-book but the whole world of observers employing such methods become more indebted to him than to the less distinguished and less talented man doing similar work; we have the assurance, not only that the book will serve its purpose, but that it represents the latest and best thought and labour of the time. This assurance is worth much to teacher and pupil; and it can hardly be doubted that the use of this little treatise will extend beyond the limits of the United States Naval Academy, where it was originally intended by its author to be used, in his own classes.

*The Analyst*, founded and edited by Joel E Hendricks, was published between 1874 and 1883. Johnson published many notes and minor papers in

*The Analyst*including:

*Bipolar Equations-Cartesian Ovals*(1875);

*The Peaucellier Machine and Other Linkages*(1875);

*Paradox for Students in Analytical Geometry*(1875);

*On the Distribution of Primes*(1875);

*Theory of Parallels*(1876);

*On the Expression 0*(1876);

^{0}*Recent Results in the Study of Linkages*(1876);

*Recent Results in the Study of Linkages [Continued]*(1876);

*Note on Evaluation of Indeterminate Forms*(1877);

*Classification of Plane Curves with Reference to Inversion*(1877);

*Singular Solutions of Differential Equations of the First Order*(1877);

*Pedal Curves*(1877);

*Symmetrical Functions of the Sines of the Angles Included in the Expression*

*a*

_{0}+ 2

*k*

^{π}/

_{n}(1879);

*Note on the Catenary*(1879);

*New Notation for Anharmonic Ratios*(1882);

*Note on Anharmonic Ratios*(1883); and

*Circular Coordinates*(1883). When

*The Analyst*ceased publication and was replaced by the

*Annals of Mathematics*, Johnson continued to publish there with papers such as:

*James Glaisher's factor tables and the distribution of primes*(1884);

*The kinematical method of tangents*(1885);

*On Singular Solutions of Differential Equations of the First Order*(1887);

*On singular solutions of differential equations of the first order*(1887);

*On the differential equation*(1887);

*On Monge's solution of the non-integrable equation between three variables*(1888); and

*On Gauss's method of substitution*(1892).

Johnson also published in the *American Journal of Mathematics*. His 1879 paper with William Story *Notes on the "15" Puzzle* is in two parts, the first by Johnson shows that the 15 puzzle is impossible using the odd/even permutation argument. He then gives a proof for non-mathematicians that:-

The second part of the paper by Story gives a generalisation to the... a permutation derived from a given one by an odd number of interchanges can never be produced by an even number of interchanges.

*n*

^{2}- 1 puzzle. Another of Johnson's papers in this journal is

*The Strophoids*(1880). He begins with a definition:-

Johnson also published several papers in theThe term Strophoid has been applied by French writers to a cubic curve, of which the symmetrical form has been discussed by Dr James Booth under the name of the Logocyclic Curve. As this curve is one of the class considered in this paper, and as the term Strophoid is appropriate to the mode of generation of the whole class, I have ventured to use the word in a more extended signification, and define the strophoid as the 'locus of the intersection of two straight lines which rotate uniformly about two fixed points in a plane'.

*Bulletin*of the American Mathematical Society.

In 1913 Johnson was given a commissioned rank in the Navy, by a special act of Congress, and when he retired in 1921 he had the rank of Commodore. After he retired he went to Johns Hopkins University. His death in 1927 was the result of bronchopneumonia. He was buried in Woodlawn Cemetery, Baltimore.

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