CQ

, or QQ is = Aa or 2 AB,

,

P always bisects QQ.

The cardioide is an algebraical curve, and the equation expressing its nature is thus:

Put a = AB the diameter, x = aD perp. AB, y = DQ perp. AD; then is which is the equation of the curve.

Many properties of the cardioide may be seen in the places above cited.

CARRÉ (Lewis), was born in the year 1663, in the province of Brie in France. His father, a substantial farmer, intended him for the church. But young Carré, after going through the usual course of education for that purpose, having an utter aversion to it, he refused to enter upon that function; by which he incurred his father's displeasure. His resources being thus cut off, he was obliged to quit the university, and look out into the world for some employment. In this exigency he had the good fortune to be engaged as an amanuensis by the celebrated father Malebranche; by which he found himself transported all at once from the mazes of scholastic darkness, to the source of the most brilliant and enlightened philosophy. Under this great master he studied mathematics and the most sublime metaphysics. After seven years spent in this excellent school, M. Carré found it necessary, in order to procure himself some less precarious establishment, to teach mathematics and philosophy in Paris; but especially that philosophy which, on account of its tendency to improve our morals, he valued more than all the mathematics in the world. And accordingly his greatest care was to make geome- try serve as an introduction to his well beloved metaphysics.

Most of M. Carré's pupils were of the fair sex. The first of these, who soon perceived that his language was rather the reverse of elegant and correct, told him pleasantly that, as an acknowledgement for the pains he took to teach her philosophy, she would teach him French; and he ever after owned that her lessons were of great service to him. In general he seemed to set more value upon the genius of women than that of men.

M. Carré, although he gave the preference to metaphysics, did not neglect mathematics; and while he taught both, he took care to make himself acquainted with all the new discoveries in the latter. This was all that his constant attendance on his pupils would allow him to do, till the year 1697, when M. Varignon, so remarkable for his extreme scrupulousness in the choice of his eleves, took M. Carré to him in that station. Soon after, viz. in the year 1700, our author thinking himself bound to do something that might render him worthy of that title, published the first complete work on the Integral Calculus, under the title of “A method of measuring Surfaces and Solids, and finding their Centres of Gravity, Percussion, and Oscillation.” He afterwards discovered some errors in the work, and was candid enough to own and correct them in a subsequent edition.

In a little time M. Carré became Associate, and at length one of the Pensioners of the Academy. And as this was a sufficient establishment for one, who knew so well how to keep his desires within just bounds, he gave himself up entirely to study; and as he enjoyed the appointment of Mechanician, he applied himself more particularly to mechanics. He took also a survey of every branch relating to music; such as the doctrine of sounds, the description of musical instruments; though he despised the practice of music, as a mere sensual pleasure. Some sketches of his ingenuity and industry in this way may be seen in the Memoirs of the French Academy of Sciences. M. Carré also composed some treatises on other branches of natural philosophy, and some on mathematical subjects; all which he bequeathed to that illustrious body; though it does not appear that any of them have yet been published. It is not unlikely that he was hindered from putting the last hand to them by a train of disorders proceeding from a bad digestion, which, after harassing him during the space of five or six years, at length brought him to the grave in 1711, at 48 years of age.

His memoirs printed in the volumes of the Academy, with the years of the volumes, are as below.

1. The Rectification of Curve Lines by Tangents: 1701.

2. Solution of a problem proposed to Geometricians, &c. 1701.

3. Reflections on the Table of Equations: 1701.

4. On the Cause of the Refraction of Light: 1702.

5. Why the Tides are always augmenting from Brest to St. Malo, and diminishing along the coasts of Normandy: 1702.

6. The Number and the Names of Musical Instruments: 1702. |

7. On the Vinegar which causes small stones to roll upon an inclined plane: 1703.

8. On the Rectification &c. of the Caustics by reflection: 1703.

9. Method for the Rectification of Curves: 1704.

10. Observations on the Production of Sound: 1704.

11. On a Curve formed from a Circle: 1705.

12. On the Refraction of Musket-balls in water, and on the Resistance of that fluid: 1705.

13. Experiments on Capillary Tubes: 1705.

14. On the Proportion of Pipes to have a determinate quantity of water: 1705.

15. On the Laws of Motion: 1706.

16. On the Properties of Pendulums; with some new properties of the Parabola: 1707.

17. On the Proportion of Cylinders that their sounds may form the musical chords: 1709.

18. On the Elasticity of the Air: 1710.

19. On Catoptrics: 1710.

20. On the Monochord: in the Machines, tom. 1. with some other pieces, not mathematical.