Descartes

OK, here are some possible applications from Descartes which may demonstrate some of the usefulness of the grid app that I've been thinking of. These are some operations on lines which Descartes introduced in 1637, la Geometrie. la Geometrie - the original edition showing ... those...some of those drawings. And main thing I'm looking at right now is: Euclid's multiplication had involved natural numbers only and resulted in quantities of squares and cubes. Descartes' operations resulted in lengths of infinitely divisible lines. Also This mesolabe compass... In the modern terminology this becomes the y-axis and this becomes the x axis and... I'll give you an idea of how that might have developed. This was from Zarlinus, in 1588 he had derived something very very similar to what Descartes has done. and Descartes had mentioned him in his Compendium Musicae of December 1618 which he wrote for Beekman. Almost immediately thereafter he wrote a letter to Beekman saying he had discovered a completely new science which would provide a general solution to all possible problems involving any sort of quantity whether continuous such as a line or discrete such as a quantity of square units... and here such as the curves drawn by the new compasses...which were continuous quantities OK Now here's his original drawing converted into...towards a more modern representation in coordinate systems. You see the relationship between the lines and the square units or the rectangular units. And then demonstrate that a little bit more here. Just an expanded version with the grid underlying it. And here's Descartes' formula... And ...it works this way: BD is to BA as BE is to BC They're similar triangles so that those relationships hold - by definition. BA equals unity. Unity is... Nowadays you could say one with the real number system but in those days you didn't... When you referred to magnitudes such as lines you could refer to unity as the defining magnitude and everything what else would be a multiple of that magnitude. Equimultiple. And so BD times BC equals BE. And... that is; if this (BA) is one; (BD/BA) becomes BD. And then you multiply BC times each side and BD times BC will be equal to BE. This becomes an equal sign in ... Descartes' kind of terminology. And there's an issue there between Euclid's definition... it would not...Euclid would never have put an equal sign (in place of the :: proportion sign) That's all I can say at this moment... This article from Ivor Grattan-Guinness indicates what the difficulty is with using an equals between ratios and also multiplying magnitudes in Euclid. Euclid would allow this. This is an... area...number...a quantity of units. Unit Area. And you can get a greatest common denominator of that which would be five... five square units. In other words a five square would be one unit. It would be the greatest common denominator of the area. And you could reapportion the area. Now you've got three units of square units. The greatest common denominator of 30 which is this length and 20 which is this length... equals three linear units and the relationship between square units and linear units is important in calculus. I don't know whether this is an instance of the fundamental theorem of calculus but it seems interesting. Seems along those lines. A little further research is needed. Here's an application to a rational number And here's an application to whole numbers. OK and that's as far as we need to get... Oh! and there's one more thing. This is the original flash... platform and on each side I've put some supporting documentation. So I have it readily available this way. Here's the screen which you just saw...we just played through. And here's the time line you can see the change. Let me bring this up here. And that's how you do it. Now here is something from Palmieri... He talks about the theory... a change between the theory of proportion and the equations. It's very important but I don't know quite what to do with this. Basically saying equimultiple proportions become problematic outside the application... outside the domain of pure mathematics...there are no simple geometrical relationships which can guarantee the principle whereby equimultiples can be associated to one another... ...I'm just not sure I don't have enough science really to say...So everything I say here has to be limited by this...(limitation). Here's...Galileo's first attempt to ... do a theory of proportion and apply it to physical dimensions. And here's...his... Palimeri's criticism of that. OK And that's that! Let me "show all" here and see if there is anything else you would like... I would like to show you. ...nope but the on any of these you can...I have the URL's available to support what I've said here. OK