Treaty of the caster

The Traité Caster is a work written in 1659 by Blaise Pascal, under the pen name Dettonville.

In 1658, Pascal is 35 years old, and already gave up making a scientific career since 1654. Nevertheless under the pseudonym of Dettonville, it will propose a challenge: to find a certain number of properties of the Cycloid .

As incredible as that can appear, it did not make a bibliography on the subject! What is discourteous with respect to Roberval.

After Wren in 1658 had carried out the correction of the cycloid one, Pascal will very quickly publish the challenges of October 1658; then the book: the Theory of the Caster, another name of cycloid (in January 1659).

This work is regarded as one of the last treaties of Geometry of Indivisible, intermediary between the Méthode of indivisible the and the calculus created by Newton (theory of the fluxions, 1669) and by Leibniz, in its more modern form (1684). This Treaty looks further into still a little the work of a Torricelli (of which it with the clarity of expression). But already Wallis (analysis infinitorum 1654), Barrow (teacher of Newton in 1661), Wren (founder of Royal Society in 1660) is on the same way and Gregory will return from Bologna (1664-1668).

This small Treaty will have the same fate as work (1641-1647) of Torricelli (death in 1647):

Indeed, Pascal falls seriously sick in 1659; jansénisme is condemned in 1661; his/her Jacqueline sister dies at the end of 1661; and Pascal, after a last invention (the first line the bus S (1662)), dies.

The Geometry of Indivisible will yield the place to the analysis. But the combinative ingeniousness of this Treaty of the Caster charms.

Reference: Pascal, complete Works, J.Chevalier ED the Pleiad.

The Treaty of the Caster

The TR (Treated Caster) comprises 18 proposals treated in 6 +1 treaties:

1. Letter of Mr. Dettonville to Mr. de Carcavi
2. Treaty of the right-angled trilignes and their miters
3. Properties of the simple sums
4. Treaty of the sines (of the quadrant)
5. Treaty of the arcs of circle
6. Small treaty of the circular solids

7. General treaty of the Caster

They will be named (Ti), I = 1..7. The 6 first make it possible to make calculations (not made by Pascal) which will be useful in (T7).

The 18 proposals are the nine problems of June and the nine of October:

That is to say an half-arch of caster of Arc OS (with O (0; 0); S (Pi.a; 2a)). H the point (Pi.a; 0), which thus forms the right-angled triligne.

Surface OSH was evaluated by Galileo (1592) (by weighing!) : 3/2 generating discs, finally calculated by Roberval (1634) using famous the " curve auxiliaire" (there = sin X, known as in modern terms); found by Torricelli then (cf the very neat investigation of Jean Itard, the Koyré center, into anteriority the Roberval one).

Pascal cuts the figure of a horizontal feature, on the basis of the point running P, cutting the half-circle of diameter HS in M, and segment HS in Y, the y-axis in Y'. Of course, PM = arc ms, which is the index property of the caster.

The 9 Proposals for a June are:

Surface PYS; X & Y of the barycentre of this surface.

To make turn the figure around PY:

Volume of the generated solid; X & Y of its barycentre.

To make turn the figure around SY

Volume of the generated solid; X & Y of its barycentre.

The 9 problems of October relate to an extension due to Wren (August 1658):

Correction of arc ms; X & Y of its barycentre.

To make turn arc ms around PY, of a half-turn:

surface of generated surface; X & Y of its barycentre.

To make turn arc ms around SY, of a half-turn:

surface of generated surface; X & Y of its barycentre.

The T1,2,4 Treaties are innovators. T3,6 take again Guldin (1637?). T5 clarifies T4 via T2. Lastly, T7 articulates the whole

The T2 Treaty is regarded by Emile Picard as a masterpiece.

Analyzes of the Treaty, notations

Notation: It will be made according to primarily the references below:

C. for Costabel and
Mr. for Merker.
TR will indicate: Treaty of the Caster.

One will call nap of the indivisible ones: ß.

Squaring of the Caster

(To entice the reader),

Here the demonstration of the squaring of the half-caster, in six lines:

To take again the preceding figure.

To divide HS into an infinity of equal parts " YY" ,

and to trace the indivisible ones, segments YP = YM + MP.

(No difficulty with) Comprendre, with Cavalieri, that ß YM = surface of the half-circle: Pi.a ² /2 (: = A1)

It remains ß MP = ß SM = ß (Pi.a - SM) = 1/2 ß Pi.a = 1/2 (Pi.a) ß = 1/2 Pi.a. (2a) = 2.A1

the surface the Roberval one, that Galileo did not know to calculate, is thus 3.aire demicercle.

END of demonstration.

Problem: must one write ß YM or like Torricelli and Pascal ß YM. YY, which in the writing of Leibniz will become: $\int_H^S YM \cdot dY$ ? We will respect the TR here, by always writing equal divisions YY and the segments (the indivisible ones) will always leave Y; this in order to remain more close to the text of Dettonville.

Art to deduct

Pascal was accustomed to combining tokens and sticks, that very small, tells his/her sister.

The Triangle of Pascal is known. And the combinative one applied to the ars conjectandi.

But of the more elementary facts exist for a long time (cf the book of Conway and Guy).

the continuation of Galileo and triangular nap

That is to say a square square. To surround on the right and by lower part of 3 squares so as to form a square tiling of 4 squares. To surround by 5 squares, one obtains a square tiling of 9 squares. Then 7 squares moreover will lead to 4^2 squares, and 9 with 5^2.

One recognizes the sum from odd the 1 with 2t-1 = t^2 of Galileo, who shows himself by recurrence: t^2 + gnomon (: = 2t + 1) = (t+1) ^2.

Pascal knew certainly this result, and that which from of deduced: summon entireties = N (n+1) /2.

but he notices more: let us name the squares by letters has, the second gnomon B, etc

The sum becomes that of a table of " Poids" : = P = HAS + 3 B + 5 C + 7 D +…

One should not Pascal a long time to discover that with this symmetrical table, one can write:

" Trace" diagonal: = ß: = (+ B HAS + C + D + E)

Either le' triangular table supérieur' , Including the diagonal, or µ; P = 2µ - ß.

To notice: µ = has + 2.B + 3.C + 4.D, i.e. the moment of the aligned weights, compared to the origin (1 is the X-coordinate of the square A): it is thus simply the moment µ lever of Archimedes.

And consequently, µ/ß gives the X-coordinate of the " barycentre" :

It is the famous rule " secrète" , about which Archimedes speaks in its letter with Dosithée. (Torricelli, Magiotti and Nardi, pupils of Castelli, in discussed much):

the X-coordinate of the barycentre is the barycentre of the X-coordinates .

Pascal calls that, to make a triangular sum µ. One can shift the origin besides.

To be exerted: to apply to the center of gravity of the quarter of circumference: $x_G = \ frac {2a} {\ pi}$ .

The barycentre of the half-disc from of deduced: (2/3). {0; $\ frac {2a} {\ pi}$ }.

To check by applying the theorem of Guldin to the sphere and the ball (stated of T4, T5 & T6).

One will as be able, by looking at the table of Pythagore with same the gnomon, to find as (ß N) ² = ß N ³: it is also traditional.

pyramidal sums

Any child, with cubes, fact of the pyramids; Dettonville is not private! That is to say axis Z downwards:

One places at the dimension z= 1 the cube has, with the dimension z=2, 4 cubes B with the dimension z=3, 9 cubes C

A beautiful pyramid thus is built.

(to take colors graduated in Z produces a pretty effect; they are well-known objects structures about it, but I forgot their name: Hoop nets?)

Obviously this pyramid has as symmetry plane the plan x=y. the whole of the elements of this diagonal plan is this time, the sum moment µ (to draw it to be convinced, if not take cubes of them!). As previously Dettonville takes half of the pyramid, WITH the symmetry plane, and calls that the pyramidal sum (þ)

Dettonville thus obtains: 2. (þ) - (µ) = 1 ². To + 2 ². B + 3 ². C + 4 ². D +5 ². E,

To include/understand Pascal, “it is necessary to remain to handle these cubes, until in being convinced”.

(This is the gasoline of what Pascal wanted to write: See & To conclude . Moreover it is the title of one of its books: the spirit of geometry and art to convince. There is a " certaine" esthetic beauty, near to the style of the " parfait" cathare, beauty undoubtedly not foreign with jansénisme of Pascal).

Direct application: to take for figure the number of the letters. In deducing again $\ Sigma$ N ³.

Equal divisions, orders

One collects bilberries with a rake of approximately 16 also spaced teeth.

That is to say 16 straws diameter equal to the space period of the rake.

To spread out the sixteen straws side by side, and possibly with a small spacer of guidance, to easily relocate them while raking.

That is a very VISUAL means to calculate surfaces, Dettonville way.

Example: To cut out a disc out of paperboard. With this one, to draw an half-arch of cycloid. Y to pose the straws. To rake the straws on the left. To introduce the directing circle out of paperboard according to HS. To bring back the straws while raking on the right: the experimental proof is made: Surface = (surface of the rectangle - surface of the half-circle, A1), that is to say 3 A1. After having shown it, it SHOULD be shown; but one is already convinced. Almost all the reasoning of Dettonville is done using this comb with bilberries, which already exists, without saying it, at Torricelli: divisions are equal on the sides of the right-angled triligne. But also an exception: like Torricelli, it gives itself the right to cut out arcs in equal lengths, which is more subtle of course, and excludes the comb with bilberry.

Here thus that the indivisible ones can be divided into 32 straws of semi-diameter, etc, up to 1024 (=2^10) tiny straws, etc It any more but does not remain to conclude, once that VISUALIZED: in extreme cases N tending towards the infinite one, for 2^n apilles, it comes that the surface under the curve has = F (1), B = F (2), C = F (3), it is the sum ß: = A+B+C+…

But Dettonville goes further:

In the exact calculation of the square Table of Galileo, he says that the weight of the diagonal is NEGLIGIBLE: thus 2µ - ß = 2µ! le' LARGE PAS' has just been crossed: the terms of " will be neglected; order inférieur" : the sum of the entireties when N is very large will be ~ N ² /2

In the exact calculation of the Pyramid, the total weight will be 2þ - µ = 2þ! Thus, the sum of the cubes will be ~ n^4/4. The concept of infinite < infinite ² < infinite ³ in the polynomials has just appeared at the great day. It is the concept of ORDER of magnitude, much more important than that which one philosophizes on the infinite ones of Pascal (but very related to the morals of Pascal). The time of the paradoxes of indivisible is not finished yet (one will need Darboux to close the debate!), but " bonnes" rules make surface.

Of course, Dettonville will not write for its double sums: ßß, and for its triple sums: ßßß, but it VISUALIZED it. It would have symbolized, it had been the creator of the calculus, the more so as it included/understood integration by parts:

The horizontal or vertical calculation: integration by parts

Obviously the surface of a curvilinear triangle as that which interests us here (half-caster OSHO), one can use the principle of the straws as well vertically as horizontally: the surface ß PY .YY + ß PY'. Y' Y' = OS.SH,

but ß PY. YY = ß PX. XX thus ß PX. XX = OS.OH - ß PY'.Y' Y'

Leibniz will write it later: D (xy) = there .dx + x.dy. For the time being, Dettonville will nothing but do make use of it, with virtuosity.

Thus compared to Torricelli which is the first with speaking in homogeneous dimensions while having given a dimension to the " dx" , Dettonville advances a step moreover with its dx.dx.dx negligible in front of of X.dx.dx, X finished. And with its double or triple sums and integration by parts.

What misses with the Treaty

the TR is not a treaty of calculus . Merker evokes:

the closed character (of the Cloister)
impossibility in calculations of nap doubles to take unequal divisions.
Wallis and Wren

closed character: Dettonville shows the fact well that nap of sine (X) = sine (X - Pi/2); thus one can make as much nap of nap of nap… than one wants. It is to some extent the beauty of the circle which would have fascinated and confined Pascal. Only T2 is a little general; but nevertheless there remains centered on this problem of the cycloid one. In addition, lira the TR is a true play of track, since one gives to it just " air of the démonstration".

In the only calculation of simple nap, with the touching , that it uses, is ß (MY.MM), it clearly indicates in T 2, qu ' it is enough that subdivide them are indefinite so that arc and tangent (or twists) merge. it is ''' Leibniz ''' which will include/understand this sentence like: to calculate the ß, it is enough to know the primitive. The tangent relation and summons of Barrow is found, but especially between derived and primitive, that firmly expressed afterwards decades of calculations cousins. However, nothing of all that exists in the TR, limited to the only geometrical remark: $\ int_a^b sin X \ cdot dx = (X \ pi/2) _a^b$ , that written in modern terms. But NEVER Dettonville does not speak about analysis. There wanted to remain geometrician.

Dettonville is limited to the equal arcs: why? Undoubtedly because the combinative spirit remains DOMINATING: he prefers to calculate like Archimedes! And, in addition its triangular formulas (µ) and pyramidal (þ) are valid only with equal divisions (Mr. p 52.57 and p114). Merker announces a very pretty paradox, p131, on the paradoxes of Tacquet, listed by Gardies: one cannot obtain without ruser the surface of the half-sphere. Darboux will also raise him of the paradoxes in the surfaces of quite selected surfaces.

Pascal was not enquis work of Roberval (cf C.). Putting questions which others already answered, it is put cantilever. From where its controversies with Wallis (already very famous). But especially when Wren publishes the correction of the cycloid in August 1658, Pascal is taken of short: to face, it rather quickly will write the problems of October. But they are hardly very glorious. Admittedly, its work (and especially T2) is admirable, but Dettonville must be, according to its morals, rather bitter to be doubled.

Conversely, Wallis had to benefit from this work. What owes Wallis with the treaty of Dettonville? Prone beautiful of investigation.

Some integrals of the Treaty of the Caster

The analysis M p50-85 is of excellent didactic, for very in love with geometry. Some integrals are quoted here (in notation modern) which manages to find Dettonville without differential calculus a priori:

Fundamental exercise: That is to say EE' the touching one out of P being projected in XX': XX'.OP = EE'.PH in our notations: D (cos $\ alpha$ = - (pH. /OP) .d $\ alpha$ . It is the only time where the touching one will intervene. Then " all is clos" because the derivative second gives again opposite Rappel function: time of Pascal, the derivative does not exist!

2: That is to say P1 and P2 being projected in H1 and H2: ß PH^n .PP = (ß HP^ (n-1). HH). R: $\ int_ {P_1} ^ {P_2} sin^n \ alpha \ cdot D \ alpha = \ int_ {H_1} ^ {H_2} sin^ {(n-1)} \ alpha \ cdot dx$ , for R= 1.

3: for all triligne right-angled, integration by parts gives: $\ int_0^A y^n \ cdot dx = (n-1) \ int_0^B xy^ {(n-1} \ cdot dy$ .

but also all kinds of formulas students' rag processions in X and there of the type $\ int \ int D (F (X, there)) = 0$ .

And Dettonville noticed geometrically that side of the curvilinear integrals, that " marchait" too. And proposals XII to XV of T2 become exploits of geometry and statics: for example to calculate ß PY' ² .PP, that is to say $\ int \ phi \ sin \ phi cos^2 \ phi \ cdot D \ phi$ , only in a geometrical way. But it remains that this type of method consisting in bringing back a curve, the caster, with a succession of circles will be not exploitable in analysis.

Conclusion (?)

In this direction, this beautiful attempt will remain dead letter. Nevertheless, let us not forget that Huygens will include/understand the isochronism of cycloid at this time: Isn't writing Pascal with Huygens the catalyst, as Wren was the catalyst of the problems of October?

And Newton, remarkable geometrician-dynamicien, if it were, is not he Pascalien without the knowledge, while renonçant with differential calculus to write Principia (cf Principia and Calculus).

References

Pascal, complete works, J.Chevalier, ED the Pleiad
Pascal, works complete, J. Mesnard, ED Desclée de Brouwer
Pascal, Thoughts, ED Flammarion 1976
, treated the indivisible ones, U Paris VII, 1987
Fermat, Works, by Tannery, ED Gauthier-Villars, 1891-1922
Leibniz, differential calculus, ED Vrin, 1989

Costabel, secrecies of the caster, Rev Hist Sc, 15, (1962)
Cederom, PU Clermont-Ferrand, 1999, by Descotes & Proust: Letters of Dettonville.
DeGandt, geometry of indivisible, APMEP, fragments of history of the maths II, 65, (1987)
Merker, song of the swan of indivisible, PUFC, (2001), ISBN 2-84627-038-4
Gardies, Pascal between Eudoxe and Cantor, 1984, ED Vrin.

See too

Blaise Pascal
Method of indivisible the
Cycloid
Roberval
Huygens
Isaac Newton
Gottfried Wilhelm von Leibniz

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