Thorn, and Moritz Cantor in his monumental work, "Vorlesungen uber Geschichte der Mathematik," is due the revival of B. Jordan's fame as a mathematician. From the elaborate investigation conducted by these two eminent critics we learn that to B. Jordan we owe the invention of syncopated algebra, in which letters are used as algebraic symbols; and in his use of letters to represent any quantities which occur in analyses, is proved the fact that he was not only far in advance of his contemporaries, but was gifted with mathematical powers of the first order.

B. Jordan denotes addition by juxtaposition. The work, "De Numeris Datis," contains solutions of one hundred and fifteen problems, quite a number of which are quadratic equations involving more than one unknown quantity. In this work he shows a knowledge of proportion that fairly astonishes a modern student. Another work of rare merit is his "Algorithmus Demonstratus," which contains partial rules for the four fundamental processes. We find in this work a general use of Arabic numerals. It is divided into ten books dealing with the properties of numbers, primes, perfect numbers, polygonal numbers, ratios, powers, and progressions.

It would seem from a study of this remarkable work that B. Jordan knew the general expression for the square of any algebraic multinomial. It was first printed and published by J. Schoner at Nuremberg, in 1534, and was until quite recently attributed to Regiomontanus (1436-1476). Geometry, it would

and another on Mechanics. Prof. Curtze for many years made a study of B. Jordan's works and is the most competent scholar on this particular subject.

Moritz Cantor-see his "Vorlesungen uber Geschichte der Mathematik,' 11 Band. von. 1200-1668. Kap. xliii-xliv. Leipzig Druck und ve:lag von B. G. Teubner, 1892.

seem, possessed no slight charm for the gifted B. Jordan. His knowledge of this interesting though difficult branch of mathematics is illustrated by his works, "De Similibus Arcubis," "De Triangulis," and "De Isoperimetris." Of these the most important is "De Triangulis," which is divided into four books. The first book, besides a few definitions, contains thirteen propositions on triangles, based on Euclid's elements. The second contains seventeen propositions, mainly on the ratio of straight lines and their application to the comparison of areas of triangles. For example, one problem is to find a point inside a triangle, so that the lines joining it to the three vertices of the triangle may divide it into three equal parts. An appeal to the text, as it is found in Cantor, shows that we moderns have not in this instance improved on B. Jordan's solution. The third book contains twelve propositions on arcs, and chords of circles, and the fourth twenty-eight propositions, partly on regular polygons, and partly on miscellaneous problems in duplication and triplication.

These by no means exhaust the list of works we are able to place to the credit of B. Jordan. They are sufficient, however, for our purpose. A deeper knowledge of the subject may be obtained by consulting the works mentioned in the foot-notes.

*Doubt has been expressed as to whether the works of B. Jordan exercised any considerable influence on the development of algebra; nowithstanding the fact that for over two hundred years his works were available to students, comparatively few derived much benefit from them, and little was done to extend the bounds of arithmetic or algebra, as there defined. Now, while

*W. W. Rouse Ball. Fellow and Lecturer, Trinity College, Cambridge University, History of Mathematics.

this indifference to the works of B. Jordan is quite marked, it is to be remembered that for over two centuries subsequent to his time, philosophical and theological studies, rather than mathematical and scientific questions, engaged the master-minds of the period. Again, it is not to be forgotten that B. Jordan was a mathematician of the first order, that his works, therefore, were the product of no ordinary mind, and that for two hundred years and over no one of sufficient genius appeared who was able to take up the work as he had left it; and, again, we must not imagine that his works indicate the general standard of mathematical learning of his time, for, as we have already observed, he stood quite alone.

After the death of B. Jordan his invaluable discoveries were disused, for the reason that they were not understood; his manuscripts were scattered, and his fame was obscured by the rising lights of scholasticism. To such an extent had his name as a mathematician been forgotten that, when the mathematical renaissance did arrive, his "Algorithmus Demonstratus" was attributed to Regiomontanus; a high compliment truly, to the genius of B. Jordan. There died at Venice in the year 1510, a Franciscan friar known as Lucas di Borgo, or Lucas Paciola, whose writings in his own day enjoyed great and well deserved popularity, as they were useful and practical treatises on arithmetic, etc. His chief work, printed in Venice in 1494, is named "Summa de Arithmetica, Geometria Proporzioni e Proporzionalita," and is the first book printed on arithmetic and algebra, the "Algorithmus Demonstratus" of B. Jordan not being printed till 1534, thereby giving to the work of the Venetian friar full forty years prior introduction to the

† Cantor, Moritz, "Vorlesungen uber Geschichte der Mathematik," II Band. Kapitel lx. Leipzig, 1892.

scholars of Europe, who, unacquainted with the abler work of the medieval mathematician, gave to Paciola the credit that rightfully belonged to B. Jordan. Modern critics have closely examined it, this work of Lucas di Borgo, and we learn from them that the arithmetical portion of his Summa was taken almost wholly-even to the examplesfrom the work of Leonardo of Pisa, a contemporary of B. Jordan, and while he uses symbols in his algebraic text, it it is to be noticed that all his equations. are numerical. He did not represent known quantities by letters as Jordan did, and as is done in modern algebra. We also learn from the same source that the claims of Rudolff and Riese to B. Jordan's discovery are unworthy of serious consideration.‡

A more successful attempt to use B. Jordan's discovery without acknowledging the debt was that made by Stifel, or Stifelious (1486-1567). Stifel had originally been an Augustinian friar, but apostatized with Luther and became, in a way, a companion of his. Possessed of mathematical ability of no mean order, his brethren in apostasy were somewhat proud of him and, with their usual modesty, claimed a great deal for him. His greatest work is entitled, "Arithmetica Integra," published at Nuremberg in 1544. So pleased were his brethren with this learned work that the preface to it was written by the famous Melancthon. Stifel devotes the third book of his "Arithmetica Integra" to algebra, and this little book has been his undoing, for in it he freely uses, without the slightest acknowledgment,* the work of B. Jordan, transcribing him almost literally. His discussion therein of known and unknown algebraic quantities, and his use of A B C to represent the unknown, bear so striking an analogy to B. Jor

See Cantor.

* Cantor, Moritz, "Vorlesungen uber Geschicl.te der Mathematik," II Band. Kapitel lxii, Leipzig, 1892.

dan's treatment of the same subject that modern criticism denies him any further consideration.

Prof. W. W. Rouse Ball,† Lecturer of Trinity College, Cambridge University, in his "History of Mathematics" asks us to give Stifel credit for re-introducing B. Jordan's work to the learned world. Is not "re-introducing" a rather mild term to apply to Stifel's act?

Through the work of Stifel, then, European scholars first learned of syncopated algebra as discovered and applied by B. Jordan. But it was not till the modern era was far advanced, and Niccola Tartaglia, or the Stammerer,

History of Mathematics, London, MacMillan & Co., 1893.

had discovered his solution of cubic equations that the full value of B. Jordan's work became manifest. Time, though tardy, is now doing justice to the genius and distinguished achievement of B. Jordan, though names once honored in the history of science have had to suffer in the process.

Modern criticism has opened up for us the vision of a medieval monk, gentle of manner, gifted above his contemporaries, living in an age when kings and emperor warred against all that he held most dear and sacred, discovering in the quiet of his cloister, in the intervals between prayer and the direction of his brethren, mathematical principles of lasting and inestimable value to the advancement of science.

III.

N less than a month, tenants had been installed in the old homestead of the Reeds, while the family had entered upon the new life in Livington. Contrary to the advice of the 'Squire, who evinced a lively interest in the affairs of his young friend, Maurice would not occupy the dwelling-house attached to the saloon, but rented a flat in a respectable neighborhood. It would be less hard for his father, he thought. Besides, a certain sense that he did not stop to analyze made him desirous to keep Dorothy and the child as far away from him as possible. He found himself shrinking more and more from their society as the passage of time made him fully acquainted with the details of the business career upon which he had embarked.

At first the amount of money that lay in the drawer at the close of a day appalled him; and he handled it as a sensitive thief might pick the pockets of his sleeping victims. Legally, it was his; morally, he felt that he had not the slightest claim to it. Yet for this he had left the quiet home in the valley; for this and the things that it would buy, he had bartered peace of mind. Yet he did not regret his action. Some one must have suffered, and it was better it should be he, here in the busy life which left little room for reflection, than Dorothy, there in the lonely country where women have no diversion from their own thoughts.

He was, in a measure, self-sufficing. He had, within, a citadel to which, when

conscience clamored too loudly or utter loathing of himself and his work became overpowering, he could retire, and, closing the door on the conscious life, give himself up to the beautiful love that awaited him in the upper chamber. In reality it was an upper chamber— the attic room on the second floor of his place of business-where the man's leisure time was spent. Here were stored some of the things for which there had been no room in the little flat. With them he had fitted up a den that for another man would have been ludicrous; for him it was pathetic. The old woollen carpet that had been his mother's pride spread its faded colors on part of the floor, a border of boards showing between its edge and the wall; in one corner was an old, hair-covered folding lounge, in another, an ancient chest of drawers; scattered around, dust-covered, lost in the big room, were a few wooden chairs; before one of the curtainless windows was a stand, with a lamp; on the wall near-by was a bookcase made out of a store-box. The few volumes that stood on the shelves were what had made life beautiful for Maurice for many years and continued to make a part of it still beautiful, notwithstanding conscience and this new loathing of duty.

But loathing it did not make him unfaithful. His obligations were numerous, and the only way to meet them was the money that was paid for the liquors he had elected to sell. The saloon had always been a prosperous one, but its patronage was of an in

ferior grade, and in taking possession of it Maurice had determined that improvement here must be his first effort. Though he could not put his soul in his. work, he would strive to better that work as far as was possible. To make it clean was his first aim; to teach his patrons that he desired drinkers and not drunkards was his second object. This cost him something at first, but ultimately he gained greatly; and before a year had elapsed The Mecca-as his predecessor had classically named the place-had gained a quiet distinction. which drew to it a better class of customers. The "Ladies Entrance" had been closed; the child with the pitcher had been forbidden to enter the doors, and the intoxicated were always conducted into the street. The high standard of the saloon began to have an appreciable effect on the immediate neighborhood. The owners of adjoining property improved it, and from being rented by second-hand dealers and cheap grocers, it passed into the possession of big business concerns and professional people.

"Reed's the best thing that has struck this part of the town in years," the citizens began to observe, and the observation brought the reflection that the saloon would not be so terrible an evil if it were properly conducted. Its worst feature, they argued, was the side door, where respectable woman, alone or with escorts, could be served. This side door for women was plainly wrong. In that we thoroughly agree with these thoughtful citizens of Livington. If it is right for a respectable man to enter a saloon to drink, it is equally right for a respectable woman to do so, and we would have her to go in by the front door, also, and not by the rear entrance. If it is wrong for either to enter the saloon, then her sin is made the greater by adding deception to the act.

Maurice's prosperity was remarkable, even in such a notably good-paying business, and, in the course of time, he bought another saloon and proceeded to build up a respectable patronage for it; in the meanwhile adding those improvements to both places that tend to make palaces for the throne of Intemperance.. When these were on solid footing, he began to invest his surplus money in real estate. There was a strange fascination for him in this money-making; sometimes in the night he would wander into the street where his new row of flats stood, and pausing before it, question if it were really true that those buildings, the homes of so many people, belonged to him-Maurice Reed-to whom a thousand dollars had once seemed a fortune. They were good homes, too. Sometimes the old question that will ever present itself to the thoughtful-as to whether he had the right to own those houses while so many thousands of his brethren were homeless or must live in the deadly tenement district-would come up to torture him; and always out of such an hour he would carry the soothing thought that, at least, they were fit habitation for men, even though these men must pay to enjoy the comfort of those houses. He was, moreover, a good landlord, and if misfortune befell any of his tenants, he proved himself a friend to his family.

The transition that had been attended by so much pain for her husband had proven easy for Mrs. Reed, and the day she saw herself mistress of the little flat in the quiet street was the opening of the gate of her heaven. The small rooms which sent Maurice to his attic over the saloon and her father-in-law to the spacious Cathedral, were her idea of the coziness which home, for her, in her present evolutionary stage, implied; and as she had natural taste and was quick