Et similiter quotcumque sint '"' This may be somewhat freely translated as follows : "The problem of the locus related to three or four lines, about which he Apollonius boasts so proudly, giving no credit to the writer who has preceded him, is of this nature: If three straight lines are given in position, and if straight lines be drawn from one and the same point, making given angles with the three given lines; and if there be given the ratio of the rectangle contained by two of the lines so drawn to the square of the other, the point lies on a solid locus given in position, namely, one of the three conic sections.

It has been shown that to only two lines there corre- sponds a plane locus. But if there be given more than four lines, the point gen- erates loci not known up to the present time that is, impossible to determine by common methods , but merely called 'lines'. It is not clear what they are, or what their properties.

One of them, not the first but the most manifest, has been examined, and this has proved to be helpful.

He translates as follows : Et on n'a fait la synthèse d' aucune de ces lignes, ni montré qu'elle servit pour ces lieux, pas même pour celle qui semblerait la première et la plus indiquée. These, however, are the propositions concerning them. Again, if there be six lines, and if the solid con- tained by three of the lines bears a given ratio to the solid contained by the other three lines, the point also lies on a 'line' given in position.

But if there be more than six lines, we cannot say whether a ratio of something contained by four lines is given to that which is contained by the rest, since there is no figure of more than three dimensions. Thus the question admits of extension to any number of lines. Then, since there is always an infinite number of different points satisfying these requirements, it is also required to discover and trace the curve containing all such points.

This is permissible, however, both to say and to demonstrate generally by this kind of proportion, and in this man- ner : If from any point straight lines be drawn making given angles with straight lines given in position; and if there be given a ratio compounded of them, that is the ratio that one of the lines drawn has to one, the second has to a second, the third to a third, and so on to the given line if there be seven lines, or, if there be eight lines, of the last to a last, the point lies on the lines that are given in position.

## Oh no, there's been an error

And similarly, whatever may be the odd or even number, since these, as I have said, correspond in position to the four lines ; therefore they have not set forth any method so that a line may be known. Ou s'il y en a fîx, que le parallélépipède côpofédetroisaitla proportion donnée auec le parafle- lepipcde des trois autres. Et ainfi cete queftiou fe peuc eftendre a tout autre nombre de lignes.

Ce qui m'a donne' occafion d'effayer fî par la méthode dont ie me ièrs on peut aller aulTy loin qu'ils ont efte'. Excepte' feule- ment lorfqu'il y a neuflignes données, fi elles font toutes parallèles. Auquel cas derechef, 8c encore en 10,11,12, ou 13 hgnes on peut trouuer les poins cherchés par le moyen d'vne hgne courbe qui foit d'vn degré plus cora- pofée que les fed;ions coniques. Puisiay trouuc'auffy, que lorfqu'il ny a que trois ou quatre hgnes données, les poins cherchés fe rencontrent tous , non feulement en l'vnedes trois fedions coni- ques , mais quelquefois aulTy en la circonférence d'vu cercle , ou en vne Hgne droite.

In this case, and in the cases where there are six, seven, eight, or nine given lines, the required points can always be found by means of the geometry of solid loci,''"' that is, by using some one of the three conic sections. Here, again, there is an exception in the case of nine parallel lines. For this and the cases of ten, eleven, twelve, or thirteen given lines, the required points may be found by means of a curve of degree next higher than that of the conic sections.

Again, the case of thirteen parallel lines must be excluded, for which, as well as for the cases of fourteen, fifteen, sixteen, and seventeen lines, a curve of degree next higher than the preceding must be used ; and so on indefinitely. Next, I have found that when only three or four lines are given, the required points lie not only all on one of the conic sections but some- times on the circumference of a circle or even on a straight line. If there are nine, ten, eleven, or twelve lines, the required curve is only one degree higher than the pre- ceding, but any such curve may meet the requirements, and so on to infinity.

GEOMETRY Finally, the first and simplest curve after the conic sections is the one generated by the intersection of a parabola with a straight line in a way to be described presently. I believe that I have in this way completely accomplished what Pappus tells us the ancients sought to do, and I will try to give the demonstration in a few words, for I am already wearied by so much writing.

They thus become lines of reference or coordinate axes, and accordingly they play a very important part in the development of analytic geometry. In this con- nection we may quote as follows: "Among the predecessors of Descartes we reckon, besides Apollonius, especially Vieta, Oresme, Cavalieri, Roberval, and Fermât, the last the most distinguished in this field; but nowhere, even by Fermât, had anv attempt been made to refer several curves of difïerent orders simultane- ously to one system of coordinates, which at most possessed special significance for one of the curves.

## E is for Explore!: Spiral Triangles | Ecole: Maths | Math art, Spiral art, Geometry art

It is exactly this thing which Descartes systematically accomplished. Heath calls attention to the fact that "the essential difference between the Greek and the modern method is that the Greeks did not direct their efforts to making the fixed lines of a figure as few as possible, but rather to expressing their equations between areas in as short and simple a form as possible. Smith, History of Mathematics, Boston, , Vol. En forte que ie penfè auoir entièrement fatisfait a ceque Pappus nous dit auoir efte'chetché'en cecy par les anciens.

Ics autrcs. FIRST BOOK such that the product of certain of them is equal to the product of the rest, or at least such that these two products shall have a çiven ratio, for this condition does not make the problem any more difficult. First, I suppose the thing done, and since so many lines are confus- ing, I may simplify matters by considering one of "the given lines and one of those to be drawn as, for example, AB and BC as the prin- cipal lines, to which I shall try to refer all the others.

Produce all the other given lines to meet these two also produced if necessary pro- vided none is parallel to either of the principal lines. We may call this ratio a : d. The translation covers the first eight lines on the original page page 32 of this edition. De plus les trois angles du triangle F S Cfont donne's, 6c en fuite îa pro- 31 Excepte feulement lî elles fontparalleles joubien a la ligne AB, auquel cas le terme compofe de la quantité AT fera nul ; oubien a la li- gne C B, auquel cas celuy qui eft compofe'de la quantité" y fera nulj ainfi qu'il eft trop manifeftc pour que ie m are- fte a l'expliquer.

Puis vous voyés aufly, que multipliant plufîeurs de ces lignes l'vne par l'autre, les quantités x3cy, qui fe trouuent dans le produit, n'y peuuentauoir que chafcu- ne autant de dimenfions, qu'il y a eu deligues, al'expli- cation 32 FIRST BOOK And thus you see that, no matter how many Hues are given in posi- tion, the length of any such hne through C making given angles with these lines can always be expressed by three terms, one of which coh- sists of the unknown quantity y multiplied or divided by some known quantity ; another consisting of the unknown quantity.

This case is too simple to require further explanation. Thus, no term will be of degree higher than the second if two lines be multiplied together, nor of degree higher than the third, if there be three lines, and so on to infinity. Rabuel, p. GEOMETRY Furthermore, to determine the point C, but one condition is needed, namely, that the product of a certain number of hues shall be equal to, or what is quite as simple , shall bear a given ratio to the product of certain other lines.

Since this condition can be expressed by a single equation in two unknown quantities,'"'' we may give any value we please to either. It is obvious that when not more than five lines are given, the quantity x, which is not used to express the first of the lines can never be of degree higher than the second. This method can be used when the problem concerns six or more lines, if some of them are parallel to either AB or BC, in which case '""' That is, an indeterminate equation. Mais au contraire fi elles font tou- tes parallèles , encore que la queftion ne foit propofee qu'en cinq lignes, ce point C ne pourra ainfi eftre trou- ue', a caufe que la quantité x ne fe trouuant point en tou- te rEquation,il ne fera plus permis de prendre vne quan- tité connue pour celle qui eft nommeej' , mais ce fera elle qu'il faudra chercher.

Et encore qu'il y ait iufques a neuf lignes don- nées, pourvûqu'elles ne foient point toutes parallèles, oiî peut toufiours faire que l'Equation ne monte que iufques auquarrédequarré. LA 36 On the other hand, if the given lines are all parallel even though a question should be proposed involving only five lines, the point C can- not be found in this way.

For, since the quantity x does not occur at all in the equation, it is no longer allowable to give a knowni value to y. It is then necessary to find the value of 3'. Such equations can always be solved by means of the conic sections in a way that I shall presently explain. Briefly, it means that for every two lines introduced the equation becomes one degree higher and the curve becomes correspondingly more complex. Those which can be solved by means of straight lines and circumferences of circles are called plane problems, since the lines or curves by which they are solved have their origin in a plane.

But problems whose solutions are obtained by the use of one or more of the conic sections are called solid problems, for the surfaces of solid figures conical surfaces have to be used. There remains a third class which is called linear because other 'lines' than those I have just described, having diverse and more involved origins, are required for their construction. Such lines are the spirals, the quadratrix, the conchoid, and the cissoid, all of which have many impor- tant properties.

Encyclopedic on Dictionnaire Raisonne des Sciences, des Arts et des Metiers, par une Société de gens de lettres, mis en ordre et publiées par M. Diderot, et quant à la Partie Mathématique par M. In substance as follows : "Mechanical is a mathematical term designating a con- struction not geometric, that is, that cannot be accomplished by geometric curves. Such are constructions depending upon the quadrature of the circle.

The term, mechanical curve, was used by Descartes to designate a curve that cannot be expressed by an algebraic equation. Geome- tric. Livre Secokd.

Mais peuteftre que ce qui a empefche' les anciens Géomètres de reçe- uou: 42 It is not because the other instruments, being more complicated than the ruler and compasses, are therefore less accurate, for if this were so they would have to be excluded from mechanics, in which accuracy of construction is even more important than in geometry. Classification par âge 3 ans et plus. Cette application peut Accéder à votre connexion Internet.

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Aller au menu Aller au contenu Aller à la recherche Nous souhaitons déposer des cookies à des fins de mesure d'audience avec Google Analytics. En savoir plus Accepter Refuser. Accueil Articles Dessinons une spirale de Fibonacci. Je ne mords pas toujours. Une petite erreur: Dans le code 1 2 3 4 5 6 7. C'est d'ailleurs précisé dans l'article. TD Oh oui, my bad, je me suis trompé dans mon calcul … désolé.