<*> circle be described whose diameter AC is = a, and AD be perpendicular and equal to AC; then taking any point P in AC, joining DP, and drawing PN parallel to AD, and NO parallel to AC; and lastly taking PM = NO, the point M will be one point of the Oval sought.

In like manner the equation expresses several very pretty Ovals, among which the following 12 are some of the most remarkable. For when the equation has four real unequal roots, the given equation will denote the three following species, in fig. 1, 2, 3:

When the two less roots are equal, the three species will be expressed as in fig. 4, 5, 6, thus:

When the two less roots become imaginary, it will denote the three species as exhibited in fig. 7, 8, 9:

When the two middle roots are equal, the species will be as appears in fig. 10: when two roots are equal, and two more so, the species will be as in fig. 11: and when the two middle roots become imaginary, the species will be as appears in fig. 12:

OUGHTRED (William)

, an eminent English mathematician and divine, was born at Eton in Buckinghamshire, 1573, and educated in the school there; whence he was elected to King's-college in Cambridge in 1592, where he continued about 12 years, and became a fellow; employing his time in close application to useful studies, particularly the mathematical sciences, which he contributed greatly, by his example and ex hortation, to bring into vogue among his acquaintances there.

About 1603 he quitted the university, and was presented to the rectory of Aldbury, near Guildford in Surry, where he lived a long retired and studious life, seldom travelling so far as London once a year; his recreation being a diversity of studies: “as often, says he, as I was tired with the labours of my own profession, I have allayed that tediousness by walking in the pleasant, and more than Elysian Fields of the diverse and various parts of human learning, and not of the mathematics only.” About the year 1628 he was appointed by the earl of Arundel tutor to his son lord William Howard, in the mathematics, and his Clavis was drawn up for the use of that young nobleman. He always kept up a correspondence by letters with some of the most eminent scholars of his time, upon mathematical subjects: the originals of which were preserved, and communicated to the Royal Society, by William Jones, Esq. The chief mathematicians of that age owed much of their skill to him; and his house was always full of young gentlemen who came from all parts to receive his instruction: nor was he without invitations to settle in France, Italy, and Holland. “He was as facetious, says Mr. David Lloyd, in Greek and Latin, as solid in arithmetic, geometry, and the sphere, of all measures, music, &c; exact in his style as in his judgment; handling his tube and other instruments at 80 as steadily as others did at 30; owing this, as he said, to temperance and exercise; principling his people with plain and solid truths, as he did the world with great and useful arts; advancing new inventions in all things but religion, which he endeavoured to promote in its primitive purity, maintaining that prudence, meekness, and simplicity were the great ornaments of his life.

Notwithstanding Oughtred's great merit, being a strong royalist, he was in danger, in 1646, of a sequestration by the committee for plundering ministers; several articles being deposed and sworn against him: but upon his day of hearing, William Lilly, the famous astrologer, applied to Sir Bulstrode Whitlocke and all his old friends; who appeared so numerous in his behalf, that though the chairman and many other Presbyterian members were active against him, yet he was cleared by the majority. This is told us by Lilly himself, in the History of his own Life, where he styles Oughtred the most famous mathematician then of Europe.—He died in 1660, at 86 years of age, and was buried at Aldbury. It is said he died of a sudden ecstasy of joy, about the beginning of May, on hearing the news of the vote at Westminster, which passed for the restoration of Charles the 2d.—He left one son, whom he put apprentice to a watch-maker, and wrote a book of instructions in that art for his use.

He published several works in his life time; the principal of which are the following: