Oeuvres complètes. Tome IV. Correspondance 1662-1663

[p. 88]

N^o 995.
W. Brouncker à Christiaan Huygens.
[1662].
Appendice au No. 994.

La pièce se trouve à Leiden, coll. Huygens.
Elle est la réponse à la Lettre No. 976.

That a Bullet let fall from any two points assign'd in the Curue HC.¹⁾ descends thereon to H in the same space of time j shall presently demonstrate. But first j aske that it be granted to me That the encrease of the velocity of the same body²⁾ is alwayes in proportion to the power of the weight. As for instance, that (because the power of the weight is in proportion to the perpendicular altitude of the inclination of the plaine, as in Steuin liure 1. de la statique³⁾ proposition 19. corollaire 2.) AB and DE being the same length, and EF to BC as 1. to a. The



encrease of the velocity of a Bullet in descending BA, be to the encrease of the velocity of the same Bullet in descending ED;⁴⁾ as a to 1. and consequently that the time it descends ED be to the time it descends BA; as 1. to √a⁵⁾.

For if the encrease of the velocity upon BA, be to the encrease of the velocity upon ED; as a to 1: or as GH to KL. IG and MK being equal as representing the same space of time; then drawing NO parallel to the base, so as the triangle NOI be equal to the Δ LKM; that the Δ LKM representing the side DE, NOI may represent the side AB, and then the time of the Bullets descent upon AB to the time of the Bullets descent upon DE, will be as IO to IG. But because HG is to LK as a to 1, therefore the Δ HGI is to the Δ NOI = Δ LKM; as a to 1. And because the Δ HGI is to the Δ NOI; as the square of the side GI, to the square of the side OI, therefore the square of GI is to the square of OI; as a to 1. therefore GI is to OI; as √a, to 1; or as 1. to √1/a.

1) Consultez la planche vis-à-vis de la page.

2) Brouncker parle de l'accélération dans l'unité de temps.

3) Voir l'ouvrage cité dans la Lettre N^o. 354, note 2.

4) Lisez: √a to 1.

5) Par le signe; Brouncker sépare les deux membres d'une proportion.

[p. 89]

Now because Xj⁶⁾, is to ba; as XH is to bH: that is (putting x for the number of the sides) as x to 1. therefore the power of the weight at X is to the power of the weight at b∷ x. 1. therefore the time of its descent from X to h, being let fall at X, is to the time of its descent from b to H, being let fall at b; as √1/x to 1. now because Xj is to hg; as XH is to hH∶ that is, as x to x-1. Therefore ABC representing the space Xh, and AB the proportionable time the Bullet is descending that space found as before



to be √1/x, let a be put for BG, the time that the said Bullet being let fall at X is passing from h to f, and let y be put for CB the velocity that the Bullet hath acquired at h in its descent from X then because AB. BC∷CF. FD. that is √1/x. y∷a. ay √x. therefore DF equals ay √x. then because Xj. hg∷DF. FE⁷⁾ (by the Petition aboue mentiond, That the encrease of the velocity is in proportion to the power of the weight; or [which is the same thing] to the perpendicular altitude of the inclination of the side.) that is x. x - 1 ∷ ay √x. x - 1/x ay√x. Therefore x - 1/x ay √x = EF. therefore Δ CEF equals x - 1/x a²y√x⁸⁾ and the Parallelogram FCBG = ay. and the Δ CBA = y/2√x. But ECBG = CBA (because hf = Xh) therefore x - 1/x ay √x⁸⁾ + ay = y/2√x, therefore a² y + 2 ay √x = y. therefore a² + 2 a √x = 1. therefore a² + 2√x/x-1 a = 1/x-1. therefore a = ⁹⁾ which is therefore the time that the Bullet falling from X descends the side hf. After the same manner the time the Bullet descends the third side is found to be , and universally the time is found to be

6) Consultez la planche vis-à-vis de la page 28.

7) Brouncker représente par GE la vitesse avec laquelle un corps, dont la chute commence en X, atteint le point f.

8) Lisez: x - 1/ 2x a² y √x.

8) Lisez: x - 1/2x a² y √x.

9) La notation √∶ 2 a - 1∶ désigne la racine √(2a - 1).

[p. 90]

l being put for the number of sides descended. now the aggregate of all these times are √∶ x² + x/2∶ abating this series 1/s² + s √∶ x² + x/2 - s² + s/2∶ s being put for 1, 2, 3, 4 &c. untill it equal x. as is euident by induction thus.

if x=1. the time is √1/1 = √1.

if x=2. the times are = √3 - ½√2.

if x=3. the times are =√6-½√5-⅙√3.

if x=4. the times are =√10-½√9-⅙√7-1/12√4.

if x=5. the times are =√15-½√14-⅙√12-1/12√9-1/20√5. &c.

Then because 1/s² + s √∶ x² + x/2 - s² + s/2∶ are the ordinates of an Ellipsis diuided by a series of triangular numbers, therefore ABC representing that Ellipsis,



and ACD the series of triangular numbers, and ACEB a Parallelogram upon the same base and altitude with the ellipsis; because the series of ordinates in the Parallelogram, diuided by the series of triangular numbers respectiuely equals twice¹⁰⁾ AB (the series being infinit) or √∶ x² + x/2∶ and because if from the series of ordinates in the parallelogram ACEB be subducted the series of ordinates in the Ellipsis ACB respectiuely, there remaines the series of ordinates in the Complement of the Ellipsis BCE: therefore the series of ordinates in the Complement of the Ellipsis diuided by the series of triangular numbers respectiuely is the aggregate of all the times. But this aggregate is still the same, because x is still the same, the number of sides (for which it is put) being alwayes infinit.

Therefore the Bullet descends from all points of this Curve in the same time Quod &c.^a).

10) Il faut biffer ce mot: twice.

a) Je n'entens pas cette demonstration depuis now the aggregate of all these times are √∶ x² + x/2∶ abating this series 1/s² + s √∶ x² + x/2 - s² + l/2 ∶⁶⁾ &c.
Je voy que les temps et leur sommes sont telles qu'il les met dans l'induction, mais je ne comprens pas ce qu'il veut prouuer par cette induction, et par consequent je n'entens pas le reste de la demonstration. Je le prie donc de me l'expliquer plus clairement depuis cet endroit la, a fin que je puisse juger si elle prouue la proposition. [Chr. Huygens].

Que toute la demonstration depuis le commencement est aussi tres obscure.
L'induction n'est point demonstration.
Que c'est une nouuelle demonstration et qu'il a abandonne la premiere.
Que je ne scaurois luy envoier la demonstration de ce que j'ay dit sans l'envoier toute entiere ce qui seroit trop long parce qu'il y a plusieurs propositions. [Chr. Huygens.]

6) Huygens a mis, par erreur, un l pour un s.

6) Huygens a mis, par erreur, un l pour un s.