Note:

Although the association of algebra and geometry was proposed even by the Greeks [8, p.84], and taken up anew as a program by Vihte, no satisfying procedure had been found to merge the two disciplines into one ( unti l the development of the Cartesian plane. Thus, Descartes was not the first to attempt to develop a coordinate plane, but his method has been the one that achieved the desired goal. Both the Greeks and Egyptians had developed a numerical coordinate system (driven by its relevance to astronomy and cartography), but with little mathematical development. 

 "Hipparchus (B.C. 150) and Ptolemy (150 A.D.), to name but two, both employed a system of latitude and longitude to locate stars on the celestial sphere.

The next person to significantly advance the creation of the coordinate system was Frangois Vihte (1540-1603). In his In artem analyticem isagoge (Introduction to the Analytical Art) published in 1591, Vihte announced a program to "[bring] together the a ncient geometrical methods of Euclid, Archimedes, Apollonius, and Pappus" [1, p.268], with ancient algebraic methods to produce his logistica speciosa, a way to formulate and solve algebraic problems.

Frangois Vihte

The Key to Geometry:
A Pair of Perpendicular Lines

Deepak Kandaswamy

René Descartes (1596-1650) is primarily associated with Philosophy: his Discourse on Method and Meditations have even led him to be called the "Father of Modern Philosophy." In his most celebrated argument, Descartes attempted to prove his own existence via the now hackneyed argument, "I think therefore I am." However, it should not be forgotten that René Descartes applied his system to investigations in physics and mathematics, with real success, playing a crucial role in the development of a link betw een algebra and geometry - now known as analytic geometry, a subject defined by Webster's New World Dictionary as "the analysis of geometric structures and properties principally by algebraic operations on variables defined in terms of position coordinates." Simply put, analytic geometry translates problems of geometry into ones of algebra. Prior to the Cartesian plane and analytic geometry, most mathematicians considered (synthetic) geometry and (diophantine) algebra to be two quite different fields of study. To anyone that has taken a high school course in analytic geometry, that notion s eems ridiculous, or even incomprehensible, but to mathematicians of 500 years ago or more, solving geometric problems using the methods of algebra probably seemed equally absurd.

In fact, as will be evident later in the paper, much of our tenth grade "vocabulary" (using x2 to represent the equation of a parabola, using terms a, b, c to denote indeterminate parameters, etc...) can trace their roots directly back to the work o f René Descartes, building on the algebra of the late 16th century.

How did it transpire that someone who had more interest in determining whether or not we live in a dream world than in, for example, determining the mean and extreme ratio mathematically, come to fundamentally change not only the way we do geometry, but also the way we think about geometry? To understand the answer, it will be useful to examine the life of René Descartes and the period in which he flourished.

Descartes' father was a lawyer and judge, and his parents belonged to the noblesse de robe, the social class of lawyers, between the bourgeoisie and the nobility. As such he received and excellent education, and had the financial resources to continue hi s studies at the Jesuit College of the town of La Flhche in Anjou [9, pp. 1-2]. Men are a product of their times, and René Descartes was no exception. After hearing that Galileo Galilei, among others, both pronounced, and persuasively argued, that the sun did not revolve around the Earth, but rather vice versa, and that, in addition , the earth made a complete revolution daily, Descartes began to question whether any of the senses could be trusted as a source of information. After all, his sense of motion clearly demonstrated that the Earth is stationary, while it was "truly" rotating and moving at a great speed through space. If his senses could be wrong in regard to something so basic, was not it possible to be equally mis taken in other fundamental areas as well? Nonetheless, according to Descartes "I concluded that I might take as a gen-eral rule the principle that all things which we very clearly and obviously conceive are true: only observing, however, that there is some difficulty in rightly determining the o bjects which we distinctly conceive." Descartes held knowledge up to a very severe standard. According to Descartes, the four rules of logic were:
1.) To accept as true only those conclusions which were clearly and distinctly known to be true.
2.) To divide difficulties under examination into as many parts as possible for their better solution.
3.) To conduct thoughts in order, and to proceed step by step from the simplest and easiest to know, to more complex knowledge.
4.) In every case to take a general view so as to be sure of having omitted nothing.
[9, p.16] Because of his severe standard, Descartes' quest for underlying truths blossomed into a distinct penchant for mathematics, where proofs were just that - undeniable knowledge. Descartes' fourth standard conveys more than just a hint of the mathematician a s well as the philosopher. Often in mathematics, solving a simple problem can be trivial. However, the formulation of a general rule to solve the problem can be infinitely more useful. Descartes seems to say in his fourth rule that the general case is the one of great importance, not the specific problem. Eventually Descartes published his ideas in a little book, or appendix, titled La Géomitrie, in 1637. Descartes major contribution in this book is considered to lie in the idea of a coordinate system, allowed problems that were considered to be strictly geometric to pass over into algebra. Although the association of algebra and geometry was proposed even by the Greeks [8, p.84], and taken up anew as a program by Vihte, no satisfying procedure had been found to merge the two disciplines into one ( unti l the development of the Cartesian plane. Thus, Descartes was not the first to attempt to develop a coordinate plane, but his method has been the one that achieved the desired goal. Both the Greeks and Egyptians had developed a numerical coordinate system (driven by its relevance to astronomy and cartography), but with little mathematical development. "Hipparchus (B.C. 150) and Ptolemy (150 A.D.), to name but two, both employed a system of latitude and longitude to locate stars on the celestial sphere." [9, p.85] The Greeks even employed a system that made use of two axes at a right angle. However, nothing systematic or permanent came out of the study of specific problems using two axes as part of the solution. Heath says that "the essential difference between t he Greek and 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." [10, p.26 bottom footno te] The first real development of a geometrical coordinate system comes in the work of Apollonios of Perga (ca. 240 - ca. 174 B.C.). Apollonios of Perga, or the "Great Geometer" as he was known, wrote a book called Conics, which, among other things, introduc ed the world to the terms parabola, ellipse, and hyperbola. In his Conics, Apollonius used a system of coordinates to solve problems regarding second-order curves (conic sections). [5, p. 211] The next person to significantly advance the creation of the coordinate system was Frangois Vihte (1540-1603). In his In artem analyticem isagoge (Introduction to the Analytical Art) published in 1591, Vihte announced a program to "[bring] together the a ncient geometrical methods of Euclid, Archimedes, Apollonius, and Pappus" [1, p.268], with ancient algebraic methods to produce his logistica speciosa, a way to formulate and solve algebraic problems. Among other things, this text uses consonants to repr esent given quantities and vowels to denote unknown quantities. This led to Vihte's nickname, "The father of modern algebra." The degree of Descartes' originality remains a subject of controversy, as will be addressed at greater length below, a controversy that has persisted in the three and a half centuries since his death. In Descartes' La Géomitrie, he uses the letters a, b , c, etc., to express already known magnitudes and x, y, z, etc., for unknown ones. Later on, Descartes unveils what appears to be the birth of a fixed set of coordinate systems in a passage begining, "Let AB, AD, EF, GH, ...be any number of straight lines given in position..." [10, p.26] Smith points out here "it should be noted that t hese lines are given in position but not in length. 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 connection we may quote as follows: 'Among the p redecessors of Descartes we reckon, besides Apollonius, especially Vihte, Oresme, Cavalieri, Roberval, and Fermat, the last the most distinguished in the field; but nowhere, even by Fermat, had any attempt been made to refer several curves of different or ders simultaneously to one system of coordinates, which at most possessed special significance for one of the curves. It is exactly this thing which Descartes systematically accomplished.' [3, p.229-230] However, Scott does not agree with this assessmen t, as will be seen below. Another person who played a key role in the creation of analytic geometry was Pierre Fermat (1601 - 1665), although it is unclear whether or not Descartes knew of Fermat's work (a subject to which we shall return), Ad Locos Planos et Solidos Isagoge. In an effort to recover some of the lost proofs of Apollonius, Fermat used a system of coordinates to refer to various curves. There was a large advance in the use of the coordinate system between Apollonios and Fermat. "In [Fermat's] published works, too, there is incontrovertible evidence that he had hit upon the idea of expressing the nature of curves by means of algebraic eq uations. How clearly in fact, he had grasped the fundamental principles of analytic geometry becomes evident after a study of the opening pages of the Isagoge, the substance of which is as follows: 'Whenever two unknown quantities are found in a final equation we have a locus and the extremity of one of them describes a right angle line or a curve. The straight line is simple and unique; the curves are infinite in number and embrace the circle, par abola, ellipse, etc...'[9, p.86] Fermat goes on to list various equations of geometric interest, such as the equation of a straight line through the origin (x/y = b/d), the equation of any straight line (b/s = (a-x)/y), the equation of certain types of circle (a2-x2=y2), the equation of certain types of ellipse (a2-x2=ky2), and the equations of certain types of hyperbola (a2+x2=ky2). These formulas should leave no doubt that Fermat understood the underlying principles of analytical geometry, and helped lay the foundation for its develop ment. The ideas with which La Géomitrie had to deal, at least potentially, were of three types according to the formulation of J.F. Scott [9, pp.88-89]: 1. The employment of coordinates as a mere instrument of description 2. Algebra and geometry collaborate on single problems 3. Transference of system and structure By analyzing these individually we can see how influential they were in the development of analytic geometry, and consider more carefully which of them are actually attributable to Descartes, according to Scott. The first item, according to Scott, consti tutes the most visible connection between Descartes' work and the Cartesian plane. In La Géomitrie, Descartes uses a system of coordinates adapted to each problem. When studying multiple curves, he uses a system of lines to unify all the separate coordi nate systems into one giant system. This account clashes with the opinion of Fink and Smith, according to whom Descartes' coordinate system was set up in advance for a general set of curves, not a particular one. As far as the second point, it is the most important in Descartes' work. Using algebra to solve geometric problems greatly enhanced the flexibility of geometry. This became a legitimate way to solve a problem, and as is often found in mathematics, the m ore ways there are to approach a class of problems, the better. An example of this given at the outset in La Géomitrie was the solution of a problem of Pappus (ca. 300 A.D.), which Descartes claimed had not been completely solved by anyone [9, p.97]. In a letter to his friend Mersenne, Descartes wrote, "J'y risous un e question qui par le timoignage de Pappus n'a p{ estre trouvie par qucun des Anciens, et l'on peut dire qu'elle ne l'a p{ estre non plus par aucun des Modernes." ("I solve a problem which defeated the ancients and the moderns alike.") Pappus' problem reads, "There being three, or four, or a greater number of right lines given in position in a plane, it is first required to find the position of a point from which we can draw as many other right lines, one to each of the given lines, mak ing a known angle with it, such that the rectangle contained by two of these drawn from this point has a given proportion either to the square on the third, if there are only three, or to the rectangle contained by the other two, if there are four. Or if there are five, the product of the remaining two lines so drawn has a given proportion to the product of the remaining two and another line, and so on." [9, p. 97] Descartes originally attempted to solve this problem using pure geometry, and was unable to. This aided Descartes in his pursuit to find another method to solve the problem. Using his newly developed analytic methods, Descartes wrote in a letter to his friend that he was able to solve the problem in just five or six weeks. Unsurprisingly, Sir Isaac Newton was the first one to solve this problem using methods of pure geometry [9, p. 97]. As to the third point that Scott raises in regard to the major achievements in La Géomitrie, it appears to be rather similar to the second, and possibly not necessary. As Scott puts it, "The structure of a whole region of geometrical theory is transferre d to a region of algebraical theory, where it brings about an instructive rearrangement of the matter and raises algebraical problems which otherwise might not have imposed themselves" [9, 89]. Among the achievements of La Géomitrie, there are many methods that are still used today. Descartes proposes a method of simultaneously handling several unknown quantities at once. Also introduced is a clearer distinction between real and imaginary root s, that helped lead to modern mathematics. Scott also says, "It is a momentous liberation when Descartes throws aside the dimensional restrictions of [Vihte] and lets the arithmetical second power a2 measure a length as well as an actual square, and the arithmetical first power a measure a square as well as an actual length." [9, p.89] In La Géomitrie, Descartes views curves of degree 2n and 2n-1 as having the same complexity, and thus as closely related. Scott even claims, "Descartes notes that this number is independent to the choice of organic coordinates. In modern language it is an invariant under change of axes. Here is a first case of invariance (A celebrated later case is Relativity). When employing coordinates we are forced to make an arbitrary choice of axes and even of the type of coordinates, and in this way we impart an arbitrary element into our methods" [9, p. 90]. Scott summarized the work of Descartes under four headings: 1.) He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes its arbitrariness. 2.) Algebra makes it possible to recognize the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all. 3.) Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method. 4.) Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all. [9, pp.92-93] Much of the work that is thus accredited to René Descartes is the subject of controversy. His reputation came under attack while he was alive, attacks which have been renewed in the 350 years since his death. Even at the time of his publication of La Gi omitrie, Descartes was forced to defend himself against claims that the work was in large part derived from the work of Pierre de Fermat and Frangois Vihte. There is no doubt that Fermat compiled his work in 1629, eight years before Descartes published La Géomitrie. However, this work of Fermat did not appear in print until 1679 (posthumously, in Opera Varia), approximately thirty years after Descartes' deat h. The question then is whether or not Descartes had access to his fellow countryman's compilation prior to it being published. Fermat gave his papers to M. Despagnet around 1629, but it is unclear whether or not Despagnet circulated these works farther . Descartes did not remain silent about such allegations. He vehemently defended himself, saying even that he had nothing to learn from his contemporary mathematicians, because they were unable to solve the ancient problems. "...and in particular he [Desc artes] leaves his readers in no doubt that he did not rate the achievements of Fermat very highly." [9, p.87]

One may wonder whether maybe the opposite was true: could Fermat have "borrowed" from Descartes? This possibility can be excluded. According to Scott, who appears to be a partisan of Descartes, Fermat's letters revealed his character to be of the highest moral caliber. One may also argue that had Fermat been familiar with Descartes' work, he would likely have adopted Descartes' notation, far superior to his own. There is in any case no evidence that Fermat ever saw Descartes' work prior to its public ation, much less prior to his own work in 1629, nor were any such allegations ever made. Scott comes to the conclusion that "It seems not impossible, therefore, that Descartes and Fermat had each made considerable progress in the new methods unconscious of what had been achieved by the other." [9, p.88] He asserts that history has numerous examples of discoveries of great importance that were made simultaneously and independently. Frangois Vihte was another mathematician whom Descartes has been ac-cused of robbing. In Vihte's In Artem Analyticam Isagoge (1591), he uses a notational system to represent algebraic equations similar to the one employed by Descartes in La Géomitrie. T his has led to speculation that much of Descartes' accomplishments were merely restatements of work Vihte had done 45 years earlier. "But Descartes' clumsy cossic notation, derived in all probability from Clavius' (a 16th and 17th century teacher at the Jesuit Collegio Romano in Rome) Algebra, which he had studied while in college, indicates that he was not familiar with Vihte's work a t this point, for Vihte's notation is clearly superior, and had he been familiar with it he could not have favored that of Clavius. Descartes was obliged to rediscover these relations, to formulate the problems in his own terms, and to develop his own me ans to solving the problem, something he was to do in a way that went far beyond Vihte's pioneering work" [5, pp. 98-99]. On the other hand, had Descartes wanted to take credit for another's ideas, it is doubtful that he would have been so overt as blatan tly to copy Vihte's notation. In this regard, Descartes wrote, "As to the suggestion that what I have written could easily have been gotten from Vihte, the very fact that my treatise is hard to understand is due to my attempt to put nothing in it that I believed to be known by either him or by anyone else...I begin the rules of my algebra with what Vihte wrote at the very end of his book, De emendatione aequationum... Thus, I begin where he left off" [10, p. 10, first paragraph of footnote]. This does of course openly acknowledge fami liarity with Vihte.

One final person declared Descartes in no uncertain terms to be a plagiarist - John Wallis (1616-1703). Wallis repeatedly and very publicly said that the main principles of coordinate geometry had already been published in Artis Analyticf Praxis by Thom as Harriot (1560-1621). Wallis wrote in Algebra (1685), a treatise designed to promote the ideas of Harriot, which were first published in 1631, that "Harriot hath laid the foundation on which Des Cartes hath built the greatest part of his Algebra or Ge ometry" [9, pp. 138-139]

"Whilst there appears little doubt that Descartes did not hesitate to avail himself of the knowledge of Harriot in his treatment of equations, it is difficult to find anything in Harriot's published works to suggest that he had devoted any attention to the subject of coordinate geometry." [7, p. 117]

How René Descartes came up with the ideas presented in his La Géomitrie is unclear. What is clear is that regardless of the source of these ideas, La Géomitrie is a work of great importance that fueled the adoption of the Cartesian plane and the develop ment of analytic geometry, allowing problems of geometry to be solved by algebraic methods.

It seems only fitting to end this paper the way Descartes ended his La Géomitrie - with a little humor and more than a little arrogance. "Et i'espere que nos neueux me sgauront gri, non seulement des choses que iay icy expliquies; mais aussy de celes que iay omises volontairemen [sic], affin de leur laisser le plaisir de les inuenter." Or as David Eugene Smith and Marcia L. Latham have it: "I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery."

Scientists announced Thursday that, after decades of effort, they have succeeded in detecting gravitational waves from the violent merging of two black holes in deep space. The detection was hailed as a triumph for a controversial, exquisitely crafted, billion-dollar physics experiment and as confirmation of a key prediction of Albert Einstein's General Theory of Relativity.
It will also inaugurate a new era of astronomy in which gravitational waves are tools for studying the most mysterious and exotic objects in the universe, scientists declared at a euphoric news briefing at the National Press Club in Washington.

[A brief history of gravity, gravitational waves and LIGO]

"Ladies and gentlemen, we have detected gravitational waves. We did it!" declared David Reitze, the executive director of the Laser Interferometer Gravitational-wave Observatory (LIGO), drawing applause from an  audience that included many of the luminaries of the physics world. The briefing was watched around the world by physicists who have long waited for such a detection.
Some of the scientists gathered for the announcement had spent decades conceiving and constructing LIGO.

From 'natural place' to gravitational waves: Gravity in 90 seconds

Play Video1:34

From Aristotle to Einstein, the world's greatest minds have long theorized about gravity. Here are the highlights, and where the study of gravity is headed next. (Gillian Brockell,Joel Achenbach/TWP)

“For me, this was really my dream. It’s the golden signal for me," said Alessandra Buonanno, who started working on theoretical models of gravitational waves in 2000 and is now a professor at Germany's Max Planck Institute for Gravitational Physics.
The observatory, described as "the most precise measuring device ever built," is actually two facilities in Livingston, La., and Hanford, Wash. They were built and operated with funding from the National Science Foundation, which has spent $1.1 billion on LIGO over the course of several decades. The project is led by scientists from the California Institute of Technology and the Massachusetts Institute of Technology, and is supported by an international consortium of scientists and institutions.

[Why this famous physicist tweeting rumors about gravitational waves]

LIGO survived years of management and funding turmoil, and then finally began operations in 2002. Throughout the first observational run, lasting until 2010, the universe declined to cooperate. LIGO detected nothing.
Then came a major upgrade of the detectors. LIGO became more sensitive. On Sept. 14, the signal arrived.
Though only a "chirp," it was a clear, compelling signal of two black holes coalescing, LIGO scientists said. It lasted less than half a second, but it captured, for the very first time, the endgame of two black holes spiraling together.

A bird's eye view of LIGO Hanford's laser and vacuum equipment area, which houses the pre-stabilized laser, beam splitter, input test masses and other equipment. (Caltech/MIT/LIGO Lab)

"This was truly a scientific moonshot," Reitze said during the announcement. "I really believe that. And we did it. We landed on the moon."
These black holes were each approximately the diameter of a major metropolis. They orbited one another at a furious pace at the very end, speeding up to about 75 orbits per second — warping the space around them like a blender cranked to infinity — until finally the two black holes became one.

[This is what it looks like when a black hole tears a star apart]

The pattern of the resulting gravitational waves contained information about the nature of the black holes. Most significantly, the signal closely matched what scientists expected based on Einstein's relativity equations. The physicists knew, from supercomputer calculations and theoretical models, what gravitational waves from merging black holes ought to look like — with a rising frequency, culminating in that chirp, followed by a "ring-down" as the waves settle.
Gabriela Gonzalez, a physics professor at Louisiana State University who is the spokesperson for LIGO, revealed images of the waves picked up by the two detectors and then played an audio version of the same signal.
"Did you hear the chirp? There's a rumbling noise, and then there's a chirp," she told the Press Club audience. "That's the chirp we've been looking for."

Power recycling optic 2. (LIGO)

This cosmic chirp was picked up by both the Louisiana and Washington state detectors. It was such a strong signal that everyone knew it was either a real detection of a black hole merger, or "somebody had injected a signal into the interferometers and not properly flagged it into the data set. It turned out that fortunately that wasn’t the case,” as Reitze put it in advance of the news conference.
He said the team, knowing the checkered history of gravitational wave detections that were later discredited, took special care to have the results verified and peer-reviewed prior to the big announcement. The scientists even looked for the possible handiwork of a computer hacker, Reitze said. All reviews held up.

[Cosmic smash-up: BICEP2’s big bang discovery getting dusted by new satellite data]

The LIGO success has been a poorly kept secret in the physics world, but the scientists kept their historic paper detailing the exact results secret until Thursday morning.
"I didn't tell my wife until a few days ago," LIGO co-founder Kip Thorne, a theoretical physicist at Caltech, said amid a scrum of reporters after the announcement. He said he'd been involved with efforts to register  gravitational waves since the 1960s. "What I feel is just profound satisfaction."
There is no obvious, immediate consequence of this physics experiment, but the scientists say this opens a new window on the universe. Until now, astronomy has been almost exclusively a visual enterprise: Scientists have relied on light, visible and otherwise, to observe the cosmos. But now gravitational waves can be used as well.
Gravitational waves are the ripples in the pond of spacetime. The gravity of large objects warps space and time, or “spacetime” as physicists call it, the way a bowling ball changes the shape of a trampoline as it rolls around on it. Smaller objects will move differently as a result — like marbles spiraling toward a bowling-ball-sized dent in a trampoline instead of sitting on a flat surface.

[Einstein's General Theory of Relativity at 100: Put that in your pipe and smoke it]

These waves will be particularly useful for studying black holes (the existence of which was first implied by Einstein's theory) and other dark objects, because they'll give scientists a bright beacon to search for even when objects don't emit actual light. Mapping the abundance of black holes and frequency of their mergers could get a lot easier.

The control room of the LIGO Hanford detector site near Hanford, Wash. (Caltech/MIT/LIGO Lab)

Since they pass through matter without interacting with it, gravitational waves would come to Earth carrying undistorted information about their origin. They could also improve methods for estimating the distances to other galaxies.
LIGO scientists said they are analyzing additional data from the observational run lasting from September to early January, and that they may find other signs of black hole mergers. One candidate for such an event, picked up in October, is still being analyzed, they said.
“The geometry of spacetime gives a burp at the end of [the merger],” said Rainer Weiss, an MIT professor of physics emeritus who has labored on LIGO since the 1970s.

[European probe launched in a search for ripples in space-time]

No one had ever seen direct evidence of “binary” black holes – two black holes paired together and then merging. The Sept. 14 signal came from about 1.3 billion light years away, though that's a very approximate estimate. That places the black hole merger in very deep space; the signal that arrived in September came from an event that happened before there were any multicellular organisms on Earth.
The reason that gravitational waves have been so difficult to detect is that their effects are tinier than tiny. In fact, the signals they produce are so small that scientists struggle to remove enough background noise to confirm them.
LIGO  detects gravitational waves by looking for tiny changes in the path of a long laser beam. In each of the lab's two facilities, a laser beam is split in two and sent down two perpendicular tubes 2.5 miles long. Each arm of the beam bounces off a mirror and heads back to the starting point. If nothing interferes, these two arms recombine at the starting point and cancel each other out.
But a photodetector is waiting in case something goes wrong. If the vibration of a gravitational wave warps the path of one of the lasers, making the two beams almost infinitesimally misaligned, the laser will hit the photodetector and alert the scientists.
To catch movement that small, scientists have to filter out ambient vibrations all the time. And sometimes even seemingly perfect results can end in disappointment: To prevent false positives, LIGO has an elaborate system in place to occasionally inject ersatz signals. Only three scientists on the team know the truth in such cases, and in at least one instance their colleagues were prepared to publish the results when they finally revealed the ruse.

Fascinating photos of our solar system and beyond

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New discoveries about Jupiter’s Great Red Spot and the latest images of Pluto.

This fail-safe gave pause to many scientists when rumors about the LIGO detection began to circulate in recent months. But the team confidently confirmed that its readings were not falsely injected – it really spotted a pair of black holes.
One of the two black holes had a mass about 36 times greater than our sun. The other registered at 29 solar masses. Both were rather massive as black holes go -- 10 solar masses is more typical.
“For the first time we have a signature of the heavy black hole forming. That was a surprise,” said Vicky Kalogera, a Northwestern University astrophysicist who has been with LIGO for 15 years. “It wasn’t a vanilla-type of black hole that we had expected.”
When the two black holes came together – spiraling in gradually rather than colliding suddenly in a linear crash – the resulting black hole was not the 65 solar masses you'd expect from basic arithmetic, but only 62. The rest was converted to energy that radiated across space in a grand gravitational burp.

[BICEP2's controversy highlights the challenges for modern science]

That burp first reached the LIGO facility in Louisiana, then the one in Washington state just 7 milliseconds later. The sequence is important, as it allowed physicists to chart the black-hole collision back to somewhere in the southern sky. And the incredibly brief time delay supports something that theorists have long believed about gravitational waves: They move at the speed of light.
“This is the most direct test of our concepts of black holes,” said David Spergel, an astrophysicist at Princeton who was not part of the LIGO team.
The scientists are scrutinizing their data for signs of other violent cosmic events. LIGO's sensitivity continues to improve, and meanwhile other labs will work to catch up to their findings.
“This is such a fantastic new window into the universe – all the rules are different,” said Michael Turner, a University of Chicago cosmologist who also was not involved with the new discovery. “This is the Galileo moment of gravity waves.”

A black-hole collision sounds like a dramatic event, but it’s not really the big news for the physicists. The headline is that LIGO finally worked. Success in detecting gravitational waves is a win for Big Science and for the institutions that backed the project.
“It had a very rough beginning,” Weiss said. “The [National Science Foundation] had a tough time explaining to other people why they would back such a crazy thing.”
Einstein’s theory led to the prediction of gravitational waves, but, as Weiss noted, “Even Einstein wasn’t very sure about this.”
LIGO is still only about a third as sensitive as it is designed to be, and improvements in coming months should let it pick up signals from deeper regions of space, the scientists said.
Caltech's Thorne, who has written extensively about black holes, warped space and time travel, shot down one speculative thought about the implication of LIGO.

“I don’t think its going to bring us any closer to being able to do time travel," he said.

A brief history of gravity, gravitational waves and LIGOThis has turned into Gravity Week here in Washington. Loads of journalists (many in town for the annual meeting of the American Association for the Advancement of Science) crammed into the National Press Club on Thursday morning for a news conference to discuss an experiment called LIGO, which stands for Laser Interferometer Gravitational-Wave Observatory.

And here's the bulletin, straight from the presser: LIGO worked. Scientists announced that they have detected a powerful gravitational wave from the violent merging of two black holes roughly a billion light-years away.

[Physicists detect gravitational waves from violent black-hole merger]

As we ponder the big news, let’s mull gravity and why it’s been such a mysterious force for so long.
Gravity is invisible, as you may have noticed, and a little bit spooky, because it seems to reach across space to cause actions at a distance without any obvious underlying mechanism. What goes up must come down, but why that is so has never been obvious.
Physicists tell us there are four fundamental forces in the universe: Gravity, electromagnetism, the strong nuclear force and the weak nuclear force. Of these, gravity is the most anemic, and yet over cosmic expanses it has shaped the universe. In our solar system, it governs the planets and moons in their orbits. On Earth, it motivates the apple to fall from the tree. You can feel it in your bones.

In this artist’s conception, a binary black hole produces gravitational waves that travel outwards and carry energy away from the system. (NASA/GSFC)

Aristotle believed that an object fell to Earth because it sought its natural place. Heavier objects, Aristotle believed, fell faster; weight was an inherent property of the object.
In the late 16^th and early 17^th centuries, Galileo brought scientific experiments into the conversation, and he discovered that a heavy object and a light object actually fall at the same speed. One biographer claimed that he proved this by dropping two spheres from the Leaning Tower of Pisa, but the story may be apocryphal. (In 1971, Apollo 15 moonwalker David Scott did his own version of the experiment, dropping a geologist's hammer and a feather and showing that they hit the lunar surface simultaneously.)

[Cosmic smash-up: BICEP2’s big bang discovery getting dusted by new satellite data]

Galileo also discovered that objects always fall with constant acceleration and along a parabolic curve. “Galileo’s observation that all falling objects trace a parabola is one of the most wonderful discoveries in all of science,” physicist Lee Smolin writes in his book “Time Reborn.”
Then came Isaac Newton. In the second half of the 17^th century, he developed a universal law of gravity. He calculated that the attraction between two bodies was equal to the product of their masses divided by the square of the distance between them. This is true on Earth as well as in space. It explains the tides. It explains the motions of the planets around the sun. This is a basic law of nature, true anywhere in the universe.
But even Newton admitted that he didn’t understand the fundamental nature of this force. Newton could describe gravity mathematically, but he didn’t know how it achieved its effects.
In the early 20^th century, Albert Einstein finally came up with an explanation, and it's rather astonishing. First he grasped that gravity and acceleration are the same thing. His General Theory of Relativity, formulated in 1915, describes gravity as a consequence of the way mass curves "spacetime," the fabric of the universe. It's all geometry. Objects in motion will move through space and time on the path of least resistance. A planet will orbit a star not because it is connected to the star by some kind of invisible tether, but because the space is warped around the star.

Albert Einstein writes out an equation on the blackboard at the Carnegie Institute, Mt. Wilson Observatory, headquarters in Pasadena, Calif., in this Jan. 14, 1931 photo. (AP)

“Gravity, according to Einstein, is the warping of space and time,” Brian Greene wrote in his book “The Elegant Universe.”
The physicist John Wheeler had a famous saying: “Mass grips space by telling it how to curve, space grips mass by telling it how to move.”
Einstein's great theory has been tested and retested and has always come out on top. Most famously, the British astronomer Arthur Eddington observed a solar eclipse in May 1919 and concluded that starlight passing close to the sun was, indeed, bent in a manner consistent with Einstein's theory. Eddington's endorsement triggered global publicity for Einstein that made him a celebrity and the personification of scientific genius.

[BICEP2's controversy highlights the challenges for modern science]

One of the predictions of Einstein’s equations (though Einstein himself wasn’t ready to buy in fully) was the existence of gravitational waves – ripples in the spacetime fabric. Scientists in subsequent decades looked for such waves to no avail. The University of Maryland physicist Joseph Weber built gravitational-wave detecting devices and claimed to have discovered such waves, but his claims were disputed and ultimately discredited. But there were physicists who were not ready to give up the quest, and they ultimately persuaded the National Science Foundation to fund the creation of LIGO, which has two facilities, one in Livingston, La., and the other in Hanford, Wash.
LIGO had its detractors from the very start because it was going to be expensive and might detect nothing at all. These waves, if they existed, would be extremely subtle. It’s not like picking up the vibration from a passing truck. The gravitational waves, in theory, should contract or expand space by an almost infinitesimal amount. A detector a couple of miles long might become longer or shorter by less than the width of a subatomic particle.
Gravitational waves pass through everything and can't be directly captured. So the two LIGO facilities use a laser beam to try to deduce the passing of a gravitational wave. The beam is split in two, with each part bouncing off mirrors perched at the end of perpendicular, airless tubes about 2.5 miles long. When those cleaved beams again converge, they should align perfectly — unless some invisible gravitational waves have come trundling through the building, stretching one tube or compressing another and thereby changing the distances traveled by the beams.
One of the controversies over LIGO was simply about the name. Was it really an “observatory”? Some astronomers weren’t ready to go there. Astronomy has always been a science built around light. When astronomers talk about observing in the optical, the infrared, or with radio waves or gamma rays or X-rays, they are talking about different wavelengths of light, each creating its own visual picture of the universe.
But gravitational waves represent a new form of cosmic information. As the scientists told us today, it's a new way of seeing the universe — or, to use a better metaphor, of hearing the universe. Physicists say this is like adding sound to what we can already see.
The movie of the universe has always been spectacular, but it will be even better with sound.

Read more:

Geometry and Arithmetic are Synthetic Peter Suber, Philosophy Department, Earlham College

Contents

• Geometry is synthetic
• Arithmetic is synthetic
• Conclusion
• Notes
• Works Cited

Kant's claim that geometry and arithmetic are synthetic is the hard part of his doctrine that they are synthetic a priori. His argument is admittedly obscure. His explanation of why "7 + 5 = 12" is synthetic (B.15ff) ⁽¹⁾ needs much more detail before it will convert skeptics. Nevertheless, I believe that Kant was right that geometry and arithmetic are synthetic, and I believe that the argument for this conclusion can be freed from all obscurity. Indeed, I believe that a clear and convincing case for this position emerges from post-Kantian mathematics itself, in particular, from non-Euclidean geometry, Gödel's first incompleteness theorem, and a certain metatheorem about the consistency of non-standard mathematics.
Kant thought that separate arguments were needed to show that geometry and arithmetic are synthetic. ⁽²⁾ He makes his case for geometry at B.4, B.16, B.40, and B.64, and for arithmetic at B.15, B.205 (cf. A.103, A only). I will follow him in this practice.
Only minor features of the arguments in this essay are original. I have collected, clarified, and tightened these arguments, and put them in some historical context. My argument that geometry is synthetic was first made by Leonard Nelson. My first argument that arithmetic is synthetic was first made by Irving Copi in 1949. Neither of these arguments has been discussed significantly in the literature. My second argument that arithmetic is synthetic is based on commonplaces in metamathematics, but has not to my knowledge been used before to support Kant.
The first hurdles to the argument are the definitions of "analytic" and "synthetic". I read Kant to offer two independent definitions or criteria of analyticity. The grammatical test is whether the predicate of a judgment is already contained in the concept of the subject, B.10f (cf. B.193). The logical test is whether I can extract the predicate from the subject "in accordance with the principle of contradiction," B.12 (cf. B.15). He elaborates this at B.190-91:

the truth of analytic judgments can always be adequately known in accordance with the principle of contradiction. The reverse of that which as concept is contained and is thought in the knowledge of the object, is always rightly denied. But since the opposite of the concept would contradict the object, the concept itself must necessarily be affirmed of it.

In short, an analytic judgment is one whose negation is a contradiction. As such, it is what logicians call a logical truth. ⁽³⁾
A synthetic judgment is any judgment which is not analytic. The other characteristic features of synthetic judgments, such as being ampliative (B.11), are consequences of their non-analyticity. ⁽⁴⁾
Kant shows no sign of realizing that he has given two criteria of analyticity rather than one, and offers no argument that the two criteria will always agree. In this paper I will use the logical test of analyticity. It is clearer than the grammatical test, easier to apply, and familiar in the areas of mathematics and metamathematics where I intend to use it.
To recap: An analytic judgment is one whose negation is a contradiction (and whose affirmation is not). ⁽⁵⁾ A synthetic judgment is non-analytic, i.e. a judgment whose affirmation and negation are both non-contradictory. I will argue that geometry and arithmetic are synthetic in this sense. I will not argue any further that this is Kant's sense of the term "synthetic".
I would like this paper to make a point about geometry and arithmetic, and also a point about Kant. But if someone produces a strong exegetical argument that Kant did not mean that analytic truths are logical truths, then I will not dispute it, but will be content to make a point only about geometry and arithmetic. ⁽⁶⁾
Geometry is synthetic
Because Kant wrote the Critique of Pure Reason (1781) before the rise of non-Euclidean geometries, and because he declared geometry to be an a priori science, he is often accused of codifying the Euclidean geometry of his day as if it were the only possible geometry. ⁽⁷⁾ Hence the advent of non-Euclidean geometry shortly after the Critique appeared, and especially Hilbert's proof in 1899 that it is as consistent as Euclidean geometry, seem to falsify Kant's account of geometry. I will argue, on the contrary, that these developments confirmed his view. At least they confirmed his view that geometry is synthetic.
First we should note the possibility that Kant was not at all surprised or "sandbagged" by the development of non-Euclidean geometries. Kant was a friend and correspondent of the Swiss mathematician, Johann Heinrich Lambert (1728-1777), one of the forerunners of non-Euclidean geometry. ⁽⁸⁾
Despite this, Kant does speak in a way which gives comfort to his critics. In many passages he suggests that the geometry he considers a priori is Euclidean. For example, at B.41:

Geometrical propositions are one and all apodeictic, that is, are bound up with the consciousness of their necessity; for instance, that space has only three dimensions.

Or again at B.204:

[Human intuition grounds such axioms as] that between two points only one straight line is possible, or that two straight lines cannot enclose a space, etc.

In these passages he might mean what his critics think he meant: that the only possible geometry is three-dimensional and Euclidean. But at least two other readings are possible, both of which preserve the possibility that mathematical theories of non-Euclidean spaces may be developed without contradiction. The first is that the space of human experience, and the space of the science of physics, are three-dimensional and Euclidean. ⁽⁹⁾ The second is that the human intuition of space depicts or describes a Euclidean space of three dimensions. ⁽¹⁰⁾ If Kant is talking about the space of physics and experience, then non-Euclidean geometry cannot refute him; only the Einsteinean physics which incorporates a non-Euclidean geometry can refute him. ⁽¹¹⁾
If Kant is talking about human intuition, then perhaps neither non-Euclidean geometry nor Einsteinean physics can refute him. ⁽¹²⁾
On the last reading, however, Kant might still be refutable. If he is making a point about human intution, or human imagination and visualizability, then it resembles one made earlier by Leibniz: that we cannot mentally picture a space of more than three dimensions. (Leibniz said we cannot picture more than three mutually perpendicular lines. ⁽¹³⁾) Eva T.H. Brann defends this view explicitly: ⁽¹⁴⁾

Any being in the least like us, in whom the capacity to image both its own thoughts and the world confronting it is central, will have an intimate geometry that is Euclidean....In Euclidean geometry alone is similarity, the preservation of shape across variations of size, possible —and similarity is the sine qua non of imaging.

As a claim about human imaging capabilities, this might well be accepted even by life-long specialists in non-Euclidean geometry and Einsteinean physics. They can do their work abstractly, representing the n dimensions of a non-Euclidean space analytically by the n arguments to a function; they need have no skill in visualizing a space of n dimensions.
What would falsify Kant on this point is not the advent of consistent non-Euclidean geometries or experimentally confirmed Einsteinean physics, but the claim made by Rudy Rucker that 15 years of diligent practice had enabled him to attain roughly 15 minutes of "direct vision of four-dimensional space." ⁽¹⁵⁾
In short, we ought to distinguish (1) the space described by a mathematical theory, from (2) the space of experience and physics, from (3) the space of human intuition —even if under some theories these are intimately related to one another. If Kant's claims were about the space of experience, physics, or human intuition, then he would not deny, even by implication, the consistency of novel geometries, qua axiom systems or mathematical theories, ⁽¹⁶⁾ although his followers did some damage by assuming that his arguments had precluded just such theories. ⁽¹⁷⁾
Kant supports these distinctions when he says, for example, that

there is no contradiction in the concept of a figure which is enclosed within two straight lines....The impossibility arises not from the concept in itself, but...from the conditions of space and of its determination. (B.268; cf. B.271)

See Alchemical Glossary below:

So now we have a basic map of creation based on the tetractys, using a combination of Pythagorean numerology, Neoplatonic cosmology, and Renaissance alchemy to describe how things come to be.  In the beginning, there is the undifferentiated, divinely simple Monad, from which all things ultimately come.  From this Monad comes a Dyad, the Differentiation between Light and Dark, or Heaven and Mundus.  From these two principles comes a Triad, the three Alchemical Reagents of Sulfur, Mercury, and Salt.  From these three reagents come the four Elements of Fire, Air, Water, and Earth.  Collectively, they form the basic building blocks, processes, and guiding principles of all creation, starting from a simple and complete One to a myriad of forms through differentiation, change, and substance.  It’s through the unification of these substances by means of alchemical processes and knowledge, or better, gnosis of principles that we reverse the process of creation to reattain Union, to ascend back to the Monad and Source.  However, these units on the tetractys tell only half the story; they’re centers and ideas unto themselves, but on their own they can’t do much.  They’re cut off from each other and don’t describe how things actually become, just that they do.

This is where the other half of the Tetractys of Life comes from: we have the stages themselves, now we just need the processes that allow things to progress or regress from stage to stage.  These paths, much as the paths on the Tree of Life in Jewish kabbalah or Hermetic qabbalah, describe how power and essence flows from one stage to the next in order to produce creation or guide henosis, and this is where we combine our meditations on the Greek alphabet with those on the tetractys of emanations.  However, in order to make use of them, we first need to actually figure out what the paths themselves should be and what spheres they should connect on the tetractys.  Also, as a side note, let’s start calling the individual units of the tetractys, when using them as alchemical/planetary concepts, as spheres.  Just to make that clear.

So, we have our unconnected Tetractys of Life.  For simplicity, I’ll use the alchemical version:

We know that all things are ultimately connected to and come from the Monad.  However, we run into a problem that’s similar to one on the Tree of Life in kabbalah; we can’t reach many of the spheres directly from the Monad since they’re blocked by other spheres.  For instance, although we might want to connect the sphere of Light with that of Fire, we must first travel through the sphere of Sulfur.  Thus, without trying to “skip” the spheres that block its way, the Monad can only connect to the two spheres immediately below it (Light and Darkness) as well as Mercury in the third row (since that way isn’t blocked by another sphere.

In fact, if we simply take an abstract tetractys and plot all possible paths on it, we end up with a set of paths like this:

While it looks nice and is as fully connected as we can get, it contains 33 paths, which is nine paths too many for our needs.  Remember, we need to aim for a set of 24, one for each of the 24 letters of the Greek alphabet, which themselves encompass all number.  Honestly, I dwelled on this for a good few days, trying to figure out how to best create a set of paths with a coherent logic; some made sense in some ways but felt jarringly off, and others just couldn’t add up to 24 even though the logic was nice.  Eventually, I settled on a way that made sense and had the right number of paths, as well as feeling “right”.

So, let’s start with something simpler than a fully connected set of paths and trying to delete off the tetractys.  First, consider the tetractys itself:

We know that, in the process of generation, the Monad (the single dot in the first row) first produces a Dyad; from a divinely simple and undifferentiated Source comes a distinction, an observer and observed, a Mind and a Word, a Creator and Creature, an active and a passive, a positive and a negative.  This gives us our first two paths on the tetractys:

Since all things contain the Monad within themselves and emulate the actions of the Monad in their own limited way, every point on the tetractys that has a lower level can be given the same two paths.  This is the same as in the generation of Plato’s Lambda (based on geometric series of 2 and 3 and their intermingled products), or Pascal’s Triangle (which is based on the addition of units).  This, then, gives us a set of 12 paths.  We’re halfway done!

Next, we assign horizontal paths between the units within the same row.  This connects the diversity into a whole order, a cosmos, within each row, reflecting the indivisibility of the Monad when differentiation and division are present in the world.  This gives us another six paths for a total of 18:

At this point, it’s hard to say what next needs to be connected.  Do we add in paths from the outermost points of the tetractys to the center?  Do we fully connect the points of the fourth row to the third?  Do we connect the second row to the fourth with vertical lines?  This is where we need to start thinking about the relationships between the things that the tetractys represents.  So, let’s take a look at what our alchemical tetractys looks like at this partially-connected stage:

One option for adding on a set of three paths would be to connect the spheres of the Monad, of Earth, and of Fire to the center sphere of Mercury.  After all, Mercury is common to all things and is present in all substance, right?  While it’s a convenient reason, it doesn’t actually hold up.  First, the Monad doesn’t actually have to connect to anything except the two spheres right below it; as an undifferentiated One, the only thing it can possibly do to become anything more complex is to become Two, after which Two can become Three.  So, the Monad shouldn’t connect to Mercury, although we might consider Mercury to be a “lower register” of the Monad in a more manifest way, being a hermaphroditic Reagent between the polarities of Salt and Sulfur.  For a similar reason, Earth and Fire shouldn’t connect to Mercury, either, since Mercury isn’t actually in them; Earth is the most passive of all the elements, being something like the caput mortuum, or the worthless remains or “dead head” without any Spirit left in them; likewise, Fire is already too hot for Mercury, and is what takes spirit out; there’s no volatile Mercury in Fire because Fire is the action of Spirit.

Another option to add a set of six paths, which would bring us up to 24, would be to connect the Monad to the fourth row to Water and Air, Earth to Light and Sulfur, and Fire to Darkness and Salt.  However, if anything, this is worse than the previous attempt, since all these forces are even more distantly connected and different from each other.  So it’d seem like the outermost points of the tetractys are too “extreme” to be connected any further than the two paths they each already have.  These nine possible paths, connecting each extreme point to the center point as well as to the two center points on far side of the tetractys, aren’t going to work for our purposes.

So, what do we know?  We know of 18 paths we should definitely have, and of nine we shouldn’t.  We have 18 + 9 = 27 paths we’ve considered so far, and 33 – 27 = 6 paths we haven’t yet.  What are those leftover six paths?

Six paths revolving around the center from the median two points on each side of the tetractys, altogether forming a hexagram.  Cute, isn’t it?  These last six paths are going to serve us well, but they’re also going to take a much different role in future analyses of the Tetractys of Life.  So, if we fully connect our alchemical Tetractys of Life with these extra six paths, we get the following:

Here, we have the median two spheres of every side of the tetractys connected to six other spheres, making these spheres all fully connected while leaving the extreme spheres connected only to their most closely associated concepts, both on their own row as well as to their neighbors.  This allows anything “middling” and not extreme (pure unmanifest, base passivity, base activity) to become anything else, including on a different level of manifestation.  For instance, Darkness is able to descend into the purely dark reagent Salt or the hermaphroditic reagent Mercury, but can also be involved in the creation of the dark (feminine) element Water as well as the active reagent Sulfur.  Darkness becoming Water makes sense, as does its corresponding opposite of Light becoming Air, since based on the structure of the tetractys we can claim that Water and Air are “lower registers” of Darkness and Light, much as Mercury is of the Monad itself.  Darkness becoming Sulfur, though?  It seems a little odd, but I claim that this works because the Three Reagents come from both the Two Principles, as a balance needs to be struck within each instead of solely by means of Mercury.  Similar reasons apply for Light being connected to Salt, Salt to Air, and Sulfur to Water.

So, with all of this done, we now have 24 paths on our Tetractys of Life.  Each path represents an ability to shift or change, a process between the stages or ingredients that the individual spheres on the Tetractys represents.  Some spheres cannot be connected to each other because of the need for intermediate stages (a path is interrupted by another sphere between the ultimate origin and destination spheres), or because a sphere is too extreme to be connected to another (a path cannot connect the outermost points of the Tetractys except to its two closest neighbors).  Now that we know of the possible means of transitioning between spheres on the Tetractys, we now have an even stronger tool at our disposal for meditation: the ability to meditatively or contemplatively explore the transitions themselves as processes of change between the stages indicated by the spheres.  While we haven’t gotten into the assignment of letters to the paths just yet, there’s still more analysis to be done on the different ways to divide the paths based on geometry and position, which will help to inform us on how to assign the letters to the paths as well as to guide us in meditation of divine concepts and divine names.

Oh, and if you’d like a more complete Tetractys of Life map that combines both alchemical and astrological symbols, here you go:

About polyphanes
I'm a software developer and Hermetic occultist living near Washington, DC, USA. I claim that I'm youthful, dashing, daring, and other things. I make things and chant stuff, and periodically write about them.

According to Paracelsus (1493-1591), the three primes or tria prima are^[1][2]

• Salt (base matter) 
• Sulfur (fluid connection between the High and the Low) 
• Mercury (omnipresent spirit of life) 

Four basic elements[edit]

Main article: Classical elements

Western alchemy makes use of the Hellenic elements. The symbols used for these are:^[1]

• Earth 
• Water 
• Air 
• Fire 

Seven planetary metals[edit]

Main articles: Classical planets in Western alchemy, Planets in astrology and Metals of antiquity

Seven metals are associated with the seven classical planets, and seven deities, all figuring heavily in alchemical symbolism. Although the metals occasionally have a glyph of their own, the planet's symbol is used most often, and the symbolic and mythological septenary is consistent with Western astrology. The planetary symbolism is limited to the seven wandering stars visible to the naked eye, and the extra-Saturnarian planets such as Neptune are not used.

• Gold dominated by Sol ☉ ☼ (  )
• Silver dominated by Luna ☽ (  )
• Copper dominated by Venus ♀ (also:  )
• Iron dominated by Mars ♂ (  )
• Tin dominated by Jupiter ♃ (  )
• Mercury (quicksilver) dominated by Mercury ☿ (  )
• Lead dominated by Saturn ♄ (  )^[1]

The Monas Hieroglyphica is an alchemical symbol devised by John Dee as a combination of the planetary metal glyphs.

Zodiac Signs

• Calcination (Aries )
• Congelation (Taurus )
• Fixation (Gemini )
• Dissolution (Cancer )
• Digestion (Leo )
• Distillation (Virgo )
• Sublimation (Libra )
• Separation (Scorpio )
• Incineration (Sagittarius )
• Fermentation (Capricorn)
• Multiplication (Aquarius )
• Projection (Pisces )

"Squaring the circle": an alchemical glyph (17th century) of the creation of the philosopher's stone