small sides, and demonstrated the
relations of these sides to the curve and its ordinates. His work,
entitled "Lectiones Geometricae," appeared in 1669. To his high
abilities was united a simplicity 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 discoveri