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Barrow was born in London. He went to school first at Charterhouse (where he was so troublesome that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to Felstead. He completed his education at Trinity College, Cambridge; after taking his degree in 1648, he was elected to a fellowship in 1649; he then resided for a few years in college, but in 1655 he was driven out by the persecution of the Independents. He spent the next four years traveling across France, Italy and even Constantinople, and after many adventures returned to England in 1659Events May 25 Richard Cromwell resigns as Lord Protector of England following the restoration of the Long Parliament, beginning a second brief period of the republican government called the Commonwealth. 24-year war between France and Spain ends with Fren. He was ordained the next year, and appointed to the Regius ProfessorshipThe Regius Professorship of Greek is one of the oldest and most prestigious of the professorships at the University of Cambridge. The chair was founded by Henry VIII in 1540 with a stipend of £40 per year, subsequently increased in 1848 by a canonry of El of GreekThe Greek language ( /Elini'k{/) is an Indo-European language which has existed from around the 14th century BC in the Cretan inscriptions called Linear B. Mycenaean Greek of this period is distinguished from later Classical or Ancient Greek of the 8th ce at CambridgeThis article is about Cambridge, England; see also other places called Cambridge. The city of Cambridge is an old English University town and the regional centre of the county of Cambridgeshire. It lies approximately 50 miles (80 km) north of London and i. In 1662Events March 18 Short-timed experiment of the first public buses holding 8 passengers begins in Paris May 3/ May 2 Catherine of Braganza marries Charles II of England as part of the dowry, Portugal cedes Bombay and Tangier to England May 9 Samuel Pepys wi he was made professor of geometryGeometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, bu at Gresham CollegeGresham College is an unusual institution of higher learning in London, which enrols no students and grants no degrees; its lectures are free and open to the public. It was founded in 1597 by Sir Thomas Gresham who also established London's Royal Exchange, and in 1663Events July 8 Charles II of England grants John Clarke a Royal Charter to Rhode Island. July 27 The British Parliament passes the second Navigation Act requiring that all goods bound for the American colonies have to be sent in English ships from English was selected as the first occupier of the Lucasian chair at Cambridge. He resigned the latter to his pupil Newton in 1669, whose superior abilities he recognized and frankly acknowledged. For the remainder of his life he devoted himself to the study of divinity. He was appointed master of Trinity College in 1672, and held the post until his death at Cambridge.
He is described as "low in stature, lean, and of a pale complexion," slovenly in his dress, and an inveterate smoker. He was noted for his strength and courage, and once when travelling in the East he saved the ship by his own prowess from capture by pirates. A ready and caustic wit made him a favourite of Charles II, and induced the courtiers to respect even if they did not appreciate him. He wrote with a sustained and somewhat stately eloquence, and with his blameless life and scrupulous conscientiousness was an impressive personage of the time.
Statue of Isaac Barrow in the chapel of Trinity College, Cambridge
His earliest work was a complete edition of the Elements of Euclid, which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of the Data. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones Opticae et Geometricae. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of the On Conic Sections of Apollonius of Perga, and of the extant works of Archimedes and Theodosius.
In the optical lectures many problems connected with the reflexion and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflexion or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow.
The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, Hudde and Sluze were working on the lines suggested by Fermat towards the methods of the differential calculus.
FIGURE: BARROW DIAGRAM goes here
Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangent MT could be found (thus determining the point T), then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn, he got a small triangle PQR (which he called the
differential triangle, because its sides PR and PQ were the differences of the abscissae and ordinates of P and Q), so that
To find QR : RP he supposed that x, y were the co-ordinates of P, and x - e, y - a those of Q (Barrow actually used p for x and m for y, but I alter these to agree with modern practice). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point.
Barrow applied this method to the curves (i) x² (x² + y²) = r²y²;
(ii) x³ + y³ = r³;
(iii) x³ + y³ = rxy, called la galande;
(iv) y = (r - x) tan πx/2r, the quadratrix; and
(v) y = r tan πx/2r.
It will be sufficient here if I take as an illustration the simpler case of the parabola y² = px. Using the notation given above, we have for the point P, y² = px; and for the point Q, (y - a)² = p(x - e).
Subtracting we get 2ay - a² = pe. But, if a be an infinitesimal quantity, a² must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe. Hence 2ay = pe, that is, e : a = 2y : p. Therefore TP : y = e : a = 2y : p. Hence TM = 2y²/p = 2x.
This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.
Adapted from "A Short Account of the History of Mathematics" (4th edition, 1908) by W. W. Rouse Ball.
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