of character almost sublime. “Tu, autem, Domine, quantus es geometra!” was written on the title-page of his Apollonius; and in the last hour he expressed his joy, that now, in the bosom of God, he should arrive at the solution of many problems of the highest interest, without pain or weariness.  The comment of the French historian conveys a sly sarcasm on the Encyclopedists:—­“On voit au reste, par-la, que Barrow etoit un pauvre philosophe; car il croiroit en l’immortalite de l’ame, et une Divinite, autre que la nature universelle."[A]

[Footnote A:  MONTUCLA. Hist. des Math.  Part iv. liv. 1.]

The Italian Cavalleri had, before this, published his “Geometry of Indivisibles,” and fully established his theory in the “Exercitationes Mathematicae,” which appeared in 1647.  Led to these considerations by various problems of unusual difficulty proposed by the great Kepler, who appears to have introduced infinitely great and infinitely small quantities into mathematical calculations for the first time, in a tract on the measure of solids, Cavalleri enounced the principle, that all lines are composed of an infinite number of points, all surfaces of an infinite number of lines, and all solids of an infinite number of surfaces.  What this statement lacks in strict accuracy is abundantly made up in its conciseness; and when some discussion arose thereupon, it appeared that the absurdity was only seeming, and that the author himself clearly enough understood by these apparently harsh terms, infinitely small sides, areas, and sections.  Establishing the relation between these elements and their primitives, the way lay open to the Integral Calculus.  The greatest geometers of the day, Pascal, Roberval, and others, unhesitatingly adopted this method, and employed it in the abstruse researches which engaged their attention.

And now, when but the magic touch of genius was wanting to unite and harmonize these scattered elements, came Newton.  Early recognized by Dr. Barrow, that truly great and good man resigned the Mathematical Chair at Cambridge in his favor.  Twenty-seven years of age, he entered upon his duties, having been in possession of the Calculus of Fluxions since 1666, three years previously.  Why speak of all his other discoveries, known to the whole world? Animi vi prope divina, planetarum motus, figuras, cometarum semitas, Oceanique aestus, sua Mathesi lucem praeferente, primus demonstravit.  Radiorum lucis dissimilitudines, colorumque inde nascentium proprietates, quas nemo suspicatus est, pervestigavit.  So stands the record in Westminster Abbey; and in many a dusty alcove stands the “Principia,” a prouder monument perhaps, more enduring than brass or crumbling stone.  And yet, with rare modesty, such as might be considered again and again with singular advantage by many another, this great man hesitated to publish to the world his rich discoveries, wishing rather to wait for maturity and perfection.  The solicitation of Dr. Barrow, however, prevailed upon him to send forth, about this time, the “Analysis of Equations containing an Infinite Number of Terms,”—­a work which proves, incontestably, that he was in possession of the Calculus, though nowhere explaining its principles.