# Focus-Directrix Hisotry

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I am researching the history of the focus-directrix property of the conic sections. Here is what I've found so far.

Aristaeus (fl.c. 300 bce - contemporary of Euclid) was conjectured to have discussed the f-d properties in his lost "Five Books on Solid Loci".

The great Greek geometers of the Hellenistic era did not discuss f-d.

Pappus (c. 300 CE) gave the focus-directrix property of the parabola (only?) He discussed Aristaeus, apparently had the latter's now lost book before him, and maybe got the property from that.

Kepler (fl c. 1620 CE) gave the names focus and directrix.

There's an awful lot of missing geometry between Pappus and Kepler. Who first defined the f-d properties of ellipses and hyperbolas? Was the writer in the Greek, Arabic, or Latin tradition? At what point did the geometric facts Newton used and attributed to "old geometers" appear?

Can anyone help me fill in the blanks?

#### marcus

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SelfAdjoint, you most likely know more than the rest of us, so tell us more. I have some trivia to add like a hundred years after Pappus there was a greek WOMAN mathematician who (it is recorded) wrote some articles on conic sections. You probably know of her---Hypatia (fl. 430 AD)---I understand that her writings are mentioned in catalogs but none have survived.

Supposedly Apollonius of Perga (roughly 260-190 BC) was the preeminent greek geometer as regards conics, and a half dozen of his books have survived including his famous one called "Conics". But he is not known to have discovered the focus-directrix fact.

Britannica says Pappus (fl. 320 AD) discovered that fact about ANY conic. You say he proved it (only?) for parabola. Britannica is sometimes wrong, but I would be curious to know what source indicates he only did it for one type of conic.

in case anyone doesnt know and is curious, its the fact that any conic has a line and a focus such that, at any point along the curve, the ratio (e) of two distances [(point to focus)/(point to line)] is the same-----and 0<e<1 means it is an ellipse, while e=1 means it's a parabola and e>1 hyperbola.

I am researching the history of the focus-directrix property of the conic sections. Here is what I've found so far.

Aristaeus (fl.c. 300 bce - contemporary of Euclid) was conjectured to have discussed the f-d properties in his lost "Five Books on Solid Loci".

The great Greek geometers of the Hellenistic era did not discuss f-d.

Pappus (c. 300 CE) gave the focus-directrix property of the parabola (only?) He discussed Aristaeus, apparently had the latter's now lost book before him, and maybe got the property from that.

Kepler (fl c. 1620 CE) gave the names focus and directrix.

There's an awful lot of missing geometry between Pappus and Kepler. Who first defined the f-d properties of ellipses and hyperbolas? Was the writer in the Greek, Arabic, or Latin tradition? At what point did the geometric facts Newton used and attributed to "old geometers" appear?

Can anyone help me fill in the blanks?

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Hi Marcus,

I had previously thought that Pappus had discovered the focus-directrix property for all the conics. This opinion was nothing I could cite, just the result of reading here and there over many years. Recently I got interested in tracking this down. The spur was a letter in Mathematical Intelligencer where the writer expressed scorn at the seventeenth century mathematicians and physicists, especially Newton, for not mastering Apollonius' Conics. This was a propos of opinions that Newton's development of the inverse square law from the geometry of the ellipse and angular momentum (Final result, his theorem 6). was not very intuitive or elegant.

So I imediately flashed that Apollonius wouldn't be any help since the problem related to the focus of the ellipse ("...The Sun at one focus...") and Apollonius had nothing to say about this. Then I remembered that the focus-directrix leads to the polar equation for the ellipse, and I started wondering how early that could have been achieved. Hence my question.

In looking up Pappus and focus-directrix in google, I didn't find much - everybody quoting from everybody else and the stuff just a throwaway to liven up an analytical geometry course. The only site that seemed actually to have read Pappus' Synagoge carefully said "for the parabola". That wasn't much to go on, so I put in the question mark.

#### quartodeciman

Is there any significance in the fact that focus and directrix are Latin words and not Greek words, or is that just a matter of later europeanization {medieval translation from Arabic into Latin}?

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Among the things I found in my google searches was that it was Kepler who coined the words focus and directrix. He was presumably writing in latin, as he usually did.

Focus means hearth in latin, and the focus of a conic is where that curve, regarded as a mirror, concentrates light, as for a burning glass. In the case of the ellipse, which has two foci, a light placed at one will have its rays concentrated at the other.

Directrix means she who steers or directs. It's probably feminine to agree with linea (line) which would be feminine in latin.

#### marcus

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using google I found a 13page paper on Pappus
which gives brief summaries of his 7 books.
the paper is by Don Allen in math dept of Texas A&M University.
Allen teaches History of Mathematics and has a series
of papers like this on major figures in the history mathematics
So this might be helpful (but you may already have found it)

http://www.math.tamu.edu/~dallen/masters/Greek/pappus.pdf

Here's the table of content for his history of greek math (contains many such chapters including on on Apollonius of Perga)

I cant find confirmation (that I feel is reliable) for believing that
Pappus knew the focus-directrix fact about all three types of conic section.

If you are right, and he didnt, then there does appear to be
an historical puzzle to solve: when, in the period 350-1550
(or roughly death of Pappus to birth of Kepler)
when did we become aware of the fact?

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#### quartodeciman

As I recall from some source, Kepler went from copernican circles to off-center circles, to ovals, and then tried ellipses as a last-ditch choice. I reckon he began with the sun at the center of a circle that can't make up its mind about which of two values its radius should be. This is rendered in coordinates by

x = A*cos [psi]
y = B*sin [psi]

, with A and B the two bounding radius values and [psi] the angle measured up and counterclockwise from the positive x axis. So,

cos2 [psi] + sin2 [psi] =
x2/A2 + y2/B2 =
1

. This is the equation of an ellipse with one semiaxis length A and the other semiaxis length B.

At [psi] = 0, the "radius" is A;
At [psi] = [pi]/2, the "radius" is B;
At [psi] = [pi], the "radius" is A;
At [psi] = 3[pi]/2, the "radius" is B;
At [psi] = 2[pi], the "radius" is A;

and so on. In between, the "radius" is in transition between A and B.

But soon Kepler turned to polar coordinates and the focus-directrix approach must have jumped out from the page. Did he really know coordinate geometry ala Descartes at the time, or did he make up his own technique? Another question: what would he think about the cosmographic significance of that directrix?

Thanks

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If I was still where I was before we moved, I would go to the local publidc library and research the question in the wonderful Dictionary of Scientific Biography. Pappus would have a generous entry and the editions of his books would be laid out. Failing that, I suppose we could all chip in for a copy of the Synagoge, English Translation!

#### marcus

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If I was still where I was before we moved, I would go to the local publidc library and research the question in the wonderful Dictionary of Scientific Biography. Pappus would have a generous entry and the editions of his books would be laid out. Failing that, I suppose we could all chip in for a copy of the Synagoge, English Translation!
selfAdjoint, I live not too far from a library where I could probably find a copy of Pappus Synagoge in translation. But
I tend not to go on errands like that.

If it is merely a question of looking in the Synagoge (assuming the library has it) for one particular thing-----a finite question to be answered----then I could do this.

what I fear is that I could not satisfy your curiosity with a single brief trip to the library-----to find out by whom and when and how this focus-directrix fact was discovered is, I suspect, material for an academic research paper.

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Marcus, I wouldn't ask you to travel for this casual question. Here is what I would do if I had the opportunity.

If I could get a look at a translation of the Synagoge, there should be some scholarly apparatus, at least an index or table of contents. I would try to find where he discussed focus-directrix (it won't be called that - references to conics as loci might be the key). Perhaps there's a modern introduction that discusses the contents with references. Then I would scan the text to determine if he does all three cases or only the parabola.

The reason I suspect only the parabola - aside from that reference - is that sometimes mathematicians who weren't special;ists in conics limited themselves to that curve. I was surprised to see that Archimedes' Floating Bodies - Part 2, only considers paraboloids of revolution. Further research question: did anybody, ever, do the general ellipsoid by Euclidean methods? I believe there's an interaction between pitching and rolling. Could the application of areas handle this?

#### quartodeciman

Do you all have Interlibrary Loan services where you are?

------

I can't match your ravenous appetite for, enduring patience with and assiduous devotion to science paleographs. But I was under the spell of one question early in my developing mental life: Newton showed that planetary orbits are probably ellipses using the inverse square central force idea of gravitation. This led to a couple of personal decisions. Firstly, after failing to dissect an ellipse like some ancient tri/tetragonizist, I determine to scrap all that and learn some diff. and int. calculus*. Thanks to L. Hogben (Mathematics for the Million), a college outline book and Thompson (Calculus Made Easy), I acquired more and more mathematical interests, though I never saw a derivation of the orbit equation until college. Secondly, after learning from an encyclopedia entry that Newton did this marvel in a book called "Philosophiae Naturalis Principia Mathematica", I guessed that I would need to learn Latin. So I signed up for it when I entered highschool. That took on a life of its own and I ended up taking 3½ years of the stuff.

------

*Imagine my delight when I learned later in life that Newton had probably used his method of fluents and fluxions in private, then cooked up geometrical demonstrations to make his work acceptable. I also heard that Archimedes also may have done some kind of pre-Cavalieri infinitesimal calculation of his curve quadratures, then, knowing the answers, cooked up proper area exhaustion demonstrations.

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--not to mention that there are good translations of Principia, stil in print. But that's a wonderful saga of the mind. Ave frater!

#### quartodeciman

Ave! Tibi gratia!

It was an awful long time ago. {clue: concurrently, strange mesons and hyperons were showing up in bubble chamber photos, with no real explanations about for their place in the scheme of things). Most I have forgotten, but I can still do a half***ed job of declining nouns/adjectives and conjugating verbs. And, one bit of pedogogic verse is etched on my brain permanently:

By ut translate the infinitive,
Let never be this rule forgot,
Use ne for ut when there's a "not".
¤

------
*main clause verbs like rogare , mandare, monere, expetere
¤ and don't forget to use the subjunctive in the subordinate clause

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Eheu! You remember more than I do!

#### marcus

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About the focus-directrix formulation of conics, a friend of mine
got interested and inquired of Don Allen (math historian) who replied in part as follows:

<<..As far as I know, the first use of the focus directrix formulation was by Dutch mathematician Jan Witt.

Here is the information about his book...
Liber Primus (Sources and Studies in the History of...

by Johan De Witt, et al (Hardcover - June 2000)

Hardcover: 296 pages ; Dimensions (in inches): 0.75 x 9.25 x 6.25
Publisher: Springer Verlag; (June 2000)
ISBN: 0387987487

This book is an English translation of the first part of the first textbook on Analytic Geometry, written in Latin by the Dutch statesman and mathematician Jan de Witt soon after Descartes invented the subject. De Witt (1625-1672) is best known for his work in actuarial mathematics ("Calculation of the Values of Annuities as Proportions of the Rents") and for his contributions to analytic geometry, including the focus-directrix definition of conics and the use of the discriminant to distinguish among
them. In addition to the translation and annotations, this volume containsan introduction and commentary, including a discussion of the role of conics in Greek mathematics.

...Translated from the original written by Jan de Witt. Contains references to cones generated as the locus of a point moving according to mathematical laws. Also contains an extensive commentary and discussion of conics in Greek mathematics. DLC: Geometry, Analytic--Early works to 1800....>>

I am a bit puzzled because De Witt's book would have been after Kepler----you say Kepler named the focus and directrix. But in any case it doesnt hurt to pass along some info about De Witt's book I hope! Whether or not it resolves the question.

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Thank you! This shows that the discovery was late, in the generation after Kepler and Fermat, and just before Newton. The fact that discovery is asserted for Witt makes it more probable that Pappus' definition only concerned the parabola. It is startling that the Islamic mathematicians, who went deep into Hellenistic geometry, didn't make the discovery.

#### marcus

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Thank you! This shows that the discovery was late, in the generation after Kepler and Fermat, and just before Newton. The fact that discovery is asserted for Witt makes it more probable that Pappus' definition only concerned the parabola. It is startling that the Islamic mathematicians, who went deep into Hellenistic geometry, didn't make the discovery.
but maybe the book does not say this or the source could be mistaken,
I am still doubtful--being pretty much ignorant of hist of math in renaissance. I hope sometime to see dewitt's book with my own eyes and find out if he in fact claims to be the discoverer of this formulation. however it does seem to shed some light and dewitt does seem like an interesting person!

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I've been thinking about the description of what de Witt did. He uses the discriminant to tell which curve it is. Now that's a trick from analytic geometry. When you rotate the axes or refer to oblique axes the equations lose the nice simple form in which you can tell a hyperbola from an ellipse by inpection. But your friend the discriminant knows; be it positive, negative or zero it picks out the curve type.

This leads me to think he worked out the focus-directrix properties in cartesian coordinates, showing how you could derive the cartesian equations from the locus property. That would be appropriate to his overall subject and would explain his being after Kepler. Kepler would have known the description (if perhaps not the proof) in terms of synthetic geometry.

#### quartodeciman

I still believe that a problem existed regarding the significance of that directrix line and, for that matter, the near focus point. How do these things relate to the original cone and cutting plane of the original definition of the conic curves? Furthurmore, in the application to planetary motion, what is the second focus for? It might take some time for a Kepler to decide that the second focus doesn't really signify anything, given that the first focus is taken to be the location of the center of the Sun.

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I recalled and just now verified, checking the Principia, that Newton made use of the second focus in his synthetic geometry proof of the inverse square law from Kepler's first two laws. See his Theorem XI, Problem VI.

#### quartodeciman

That is interesting and strange. I guess I must go check that out.

#### quartodeciman

I got a help guide* for Newton's PNPM direct problem (what's the centripetal force?)

Newton chose to represent the planet's positions P and R on the aphelion side of the ellipse, closer to the alternate focus, rather than on the perihelion side, closer to the Sun, S. I wonder how the demonstration would change if he put it on the perihelion side instead?

As I expected, he milks the sum-of-distances-from-the-foci invariance to get the major axis length into the demonstration.

---

*Brackenridge,The Key to Newton's Dynamics (The Kepler Problem and the Principia), University of California Press(1995)

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Right. I think the conjugate diameter would be harder to draw, but the proof should still go throught if P is on the same side of the ellipse as S. We should both draw it up to see.

#### quartodeciman

The conjugate diameters should just follow the same definitions. One diameter runs through the ellipse center parallel to the tangent at P; the other diameter runs from P through the ellipse center. Newton's particular choice of P makes it look like the two diameters are supposed to be equal and make equal angles with the major axis, but that isn't generally true.

In Theorem 4 Problem 4 (elliptical orbit with sun at one focus and inverse square force on planet-> third Kepler law) the diagram puts the planet P on the perihelion side. :)

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