These pages are from the brief work *On the Transformation of Curves into Straight Lines*, by Hendrick van Heuraet (1634 - 1660), published in the 1659 Latin edition of Descartes's *Geometry*, edited by van Schooten. Although van Heuraet was not the first to accomplish a rectification, a task that Descartes had said could not be done, this is the first publication of a general procedure, a procedure very close to our standard calculus procedure for finding the length of a curve.

On these two pages, van Heuraet describes his general procedure for rectification, one which tranforms the length into an integral, that is, the area under a curve. He then illustrates the procedure by calculating the length of the semi-cubical parabola, *y*^{2} = *x*^{3}/*a. * (We can take *a* = 1 for simplicity.) Note that since the procedure for finding arc length involved first finding *dy*/*dx* (or the tangent to the curve)*,* van Heuraet accomplishes this by using Descartes's normal method and Hudde's rule for finding a double root. Note also that van Heuratet uses Descartes's symbol for "equal" rather than our modern equal sign.

Van Heuraet's Rectification of Curves
This pages is from the *Zhoubi suanjing* (*Arithmetical Classic of the Gnomon and the Circular Paths of Heaven*), a Chinese book on astronomy and mathematics dated to approximately 100 BCE. This diagram illustrates a square of side 4 fitting into a square of side 5.
Zhoubi suanjing

Euclid’s propositions I-46 and I-47 as given in Christopher Clavius’ ( 1538-1612) *Elements* published in Rome in 1574. Note that Clavius indicates his volume contains 15 books of Euclid. Many medieval authors erroneously attributed two extra books to Euclid's *Elements.*Book XIV extends Euclid discussion in book XIII on the comparison of the regular solids inscribed in a sphere. This work is now believed to have been composed by Hypsicles of Alexandria (ca.190 BCE—ca 120 BCE). Book XV also deals with the properties of regular solids and is believed to have been compiled by Isidore of Miletus (fl.ca. 532), who was the architect responsible for the Cathedral of Holy Wisdom in Constantinople, later to become the Hagia Sophia.

Christopher Clavius's edition of Euclid's Elements

This image is page 263 of *Elements of Linear Curves *by Jan de Witt (1625 - 1672). De Witt was a student of Frans van Schooten, who published this work in his 1661 edition of Descartes' *Geometry*. (This copy is from the 1683 edition. And, in fact, de Witt himself probably finished the work by 1646.) The first of the two books of this treatise was devoted to developing the properties of the conic sections using synthetic methods based on the work of Apollonius. But in the second book, de Witt produced a complete algebraic treatment of the conics, beginning with equations in two variables, based on the work of Fermat and Descartes.

On this page, de Witt shows how to rotate the axes to turn a complicated second degree equation in two variables into the standard one displayed earlier. Unlike in modern treatises, de Witt does not use trigonometry, but gives the equations of the new axes in terms of the old ones. That is, he uses a transformation of coordinates based on the form of the given equation.

More pages from Jan de Witt's Elements of Curves
This page is from the 1536 edition of the *Libro di Arithmetica i Geometria* of Francesco Feliciano (first half of 16th century). Not much is known about Feliciano, except that he was born in Lazisa, near Verona and was still living in 1563. This book is basically a revision of Feliciano's earlier *Libro de Abaco*, which appeared in 1517. The book contains much commercial arithmetic, but also a treatment of roots, the rule of false position, some algebra, and a section on practical geometry. The book had a good deal of influence on the teaching of elementary mathematics, appearing in numerous editions including one in 1669, 143 years after the original edition.

On this page, Feliciano shows how to calculate the circumference and area of a circle. Note that he approximates pi by 22/7 to calculate the circumference and area of a circle of diameter 14.
More pages from Francisco Feliciano's Libro di Arithmetica