imported>Paul Wormer |
imported>Paul Wormer |
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| '''Charles Marie de La Condamine (Paris, January 27, 1701 -Paris, February 4, 1774)''' was a son of Charles de La Condamine and Louise Marguerite Chourses. He studied at the [[Collège Louis-le-Grand]] where he was trained in humanities as well as in mathematics. After finishing his studies, he enlisted in the army and fought in the war against Spain (1719). After returning from the war, he entered scientific circles in Paris. On December 12, 1730 he was appointed Assistant Chemist at the [[Académie des Sciences]] in Paris and became a member.
| | ==Parabolic mirror== |
| | {{Image|Refl parab.png|right|350px|Fig. 2. Reflection in a parabolic mirror}} |
| | Parabolic mirrors concentrate incoming vertical light beams in their focus. We show this. |
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| The next year (May 1731) he sailed on a ship of the Levant Company to [[Constantinople]] (now [[Istanbul]]) where he stayed five months. After returning to Paris La Condamine submitted in November 1732 a paper to the Academy entitled ''Mathematical and Physical Observations made during a Visit of the Levant in 1731 and 1732''.
| | Consider in figure 2 the arbitrary vertical light beam (blue, parallel to the ''y''-axis) that enters the parabola and hits it at point ''P'' = (''x''<sub>1</sub>, ''y''<sub>1</sub>). The parabola (red) has focus in point ''F''. The incoming beam is reflected at ''P'' obeying the well-known law: incidence angle is angle of reflection. The angles involved are with the line ''APT'' which is tangent to the parabola at point ''P''. It will be shown that the reflected beam passes through ''F''. |
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| Three years later he joined an expedition to present-day Ecuador that had the aim to test a hypothesis of Newton. Newton had posed that the Earth bulges around the equator and is flattened in the polar regions. Newton's opinion had raised a huge controversy among French scholars. [[Maupertuis]], [[Clairaut]], and [[Le Monnier]] were to travel to Lapland, where they would measure the length of several degrees of longitude along the arctic circle, while [[Godin]], [[Bouguer]], and La Condamine were sent to South America to perform similar measurements along the [[equator]].
| | Clearly ∠''BPT'' = ∠''QPA'' (they are vertically opposite angles). Further ∠''APQ'' = ∠''FPA'' because the triangles ''FPA'' and ''QPA'' are congruent and hence ∠''FPA'' = ∠''BPT''. |
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| On May 16, 1735, La Condamine sailed from La Rochelle accompanied by Godin, Bougier, and a botanist [[Joseph de Jussieu]]. After stops in [[Martinique]], [[Santo Domingo]], [[Cartagena (Columbia)]], and Panama, where the crossed land. They arrived finally (March 10, 1736) at the Pacific Port of [[Manta]] in the province of [[Quito]]. From Manta, Condamine took a route separate from Godin and Bouguer and joined them again in Quito on June 4, 1736.
| | We prove the congruence of the triangles: By the definition of the parabola the line segments ''FP'' and ''QP'' are of equal length, because the length of the latter segment is the distance of ''P'' to the directrix and the length of ''FP'' is the distance of ''P'' to the focus. The point ''F'' has the coordinates (0,''f'') and the point ''Q'' has the coordinates (''x''<sub>1</sub>, −''f''). The line segment ''FQ'' has the equation |
| | | :<math> |
| The longitudinal arc that was chosen passed through a high valley perpendicular to the equator, stretching from Quito in the north to [[Cuenca]] in the south. The scientists spent a month performing triangulation measurements in the Yaruqui plains, from October 3 to November 3, 1736. December of that year they returned to the capital of the province, Quito. After they arrived there, they found that subsidies expected from Paris had not come in. La Condamine, who had taken precautions and had made in advance a deposit on a bank in [[Lima]], travels early 1737 to Lima to collect his money. He prolonged his journey to observe the [[Cinchona tree]] with its medicinally active bark, that was little known to Europeans.
| | \lambda\begin{pmatrix}0\\ f\end{pmatrix} + (1-\lambda)\begin{pmatrix}x_1\\ -f\end{pmatrix}, \quad 0\le\lambda\le 1. |
| | | </math> |
| La Condamine returning to Quito on June 20, 1737, finds that Godin refuses to disclose his findings, and consequently he joins forces with Bouguer. They continue making length measurements in the mountainous and difficult region north of Quito. When in December 1741 Bouguer, checking a calculation of La Condamine detects an error, also these two explorers get into a quarrel and stop speaking to each other. The two men, working separately, complete their project in May 1743.
| | The midpoint ''A'' of ''FQ'' has coordinates (λ = ½): |
| | | :<math> |
| La Condamine chooses to return by way of the Amazon River, a route which is longer and more dangerous. He reaches the Atlantic Ocean at [[Para]] on September 19, 1743, having made on the way observations of astronomic and topographic interest. He also studied Cinchona and Rubber trees. In February 1774 we find him in [[Cayenne]], the capital of French Guiana. Finding no passage to France, he waits there for five months making many observations of physical, biological, and ethnological nature. Finally leaving Cayenne in August 1744 he arrives in Amsterdam on November 30, 1744 from where two months to travel to Paris where he arrives in February 1745. He brought with him many notes, natural history specimens, and art objects that he donates to the naturalist [[Georges-Louis Leclerc, Comte de Buffon|Buffon]] (1707–1788).
| | \frac{1}{2}\begin{pmatrix}0\\ f\end{pmatrix} + \frac{1}{2}\begin{pmatrix}x_1\\ -f\end{pmatrix} = |
| | | \begin{pmatrix}\frac{1}{2} x_1\\ 0\end{pmatrix}. |
| The scientific results of the expedition are clear, the Earth is indeed a spheroid flattened at the poles as was believed by Newton. Not unsurprisingly, La Condamine and Bouguer fail to write a joint publication. Only Bouguer's death in 1758 put an end to their quarrel. Godin died in 1760. | | </math> |
| | | Hence ''A'' lies on the ''x''-axis. |
| The surviving member La Condamine, helped by his writing gifts, obtained most of the credits for the expedition, although he had less talent in astronomy than Godin and was a lesser mathematician than Bouguer.
| | The parabola has equation, |
| | | :<math> |
| La Condamine had contracted smallpox in his youth. This led him to take part in the debate on vaccination against the disease and to propagate its use. Helped by the clarity and elegance of his writing, he presented several papers at the Academy of Sciences in which he defended his ideas with passion.
| | y = \frac{1}{4f} x^2. |
| | | </math> |
| He became a corresponding member of the academies of London, Berlin, St. Petersburg and Bologna and was elected to the l'Académie française on November 29, 1760.
| | The equation of the tangent at ''P'' is |
| | | :<math> |
| In August 1756, he married with papal dispensation his young niece, Charlotte Bouzia of Estouilly. La Condamine had many friends, the closest being Maupertuis whom he bequeathed his papers. Condamine died at Paris February 4, 1774, following a hernia operation.
| | y = y_1 + \frac{x_1}{2f} (x-x_1)\quad \hbox{with}\quad y_1 = \frac{x_1^2}{4f}. |
| | </math> |
| | This line intersects the ''x''-axis at ''y'' = 0, |
| | :<math> |
| | 0 = \frac{x_1^2}{4f} - \frac{x_1^2}{2f} + \frac{x_1}{2f} x |
| | \Longrightarrow \frac{x_1}{2f} x = \frac{x_1^2}{4f} \longrightarrow x = \tfrac{1}{2}x_1. |
| | </math> |
| | The intersection of the tangent with the ''x''-axis is the point ''A'' = (½''x''<sub>1</sub>, 0) that lies on the midpoint of ''FQ''. The corresponding sides of the triangles ''FPA'' and ''QPA'' are of equal length and hence the triangles are congruent. |
Parabolic mirror
PD Image Fig. 2. Reflection in a parabolic mirror
Parabolic mirrors concentrate incoming vertical light beams in their focus. We show this.
Consider in figure 2 the arbitrary vertical light beam (blue, parallel to the y-axis) that enters the parabola and hits it at point P = (x1, y1). The parabola (red) has focus in point F. The incoming beam is reflected at P obeying the well-known law: incidence angle is angle of reflection. The angles involved are with the line APT which is tangent to the parabola at point P. It will be shown that the reflected beam passes through F.
Clearly ∠BPT = ∠QPA (they are vertically opposite angles). Further ∠APQ = ∠FPA because the triangles FPA and QPA are congruent and hence ∠FPA = ∠BPT.
We prove the congruence of the triangles: By the definition of the parabola the line segments FP and QP are of equal length, because the length of the latter segment is the distance of P to the directrix and the length of FP is the distance of P to the focus. The point F has the coordinates (0,f) and the point Q has the coordinates (x1, −f). The line segment FQ has the equation
The midpoint A of FQ has coordinates (λ = ½):
Hence A lies on the x-axis.
The parabola has equation,
The equation of the tangent at P is
This line intersects the x-axis at y = 0,
The intersection of the tangent with the x-axis is the point A = (½x1, 0) that lies on the midpoint of FQ. The corresponding sides of the triangles FPA and QPA are of equal length and hence the triangles are congruent.
