Hipparchus: The Trigonometry of our Cosmos

Gabrielle Birchak/ August 23, 2022/ Ancient History, Classical Antiquity, Uncategorized

Transcript

Space. It’s not just the final fron­tier. It’s not just an infi­nite three-dimen­sion­al plat­form where enti­ties have posi­tion and direc­tion. Space is a pro­found reminder that we are part of this giant con­struct of atoms, mol­e­cules, ele­ments, com­pounds, voids, mass­es, and grav­i­ty, that work togeth­er as a uni­fy­ing body that moves us in the universe’s dance of life! Start­ing with the Big Bang, we began to exist, even though we were in the dust scat­tered across the inflat­ing universe.

Many of us exam­ine the stars to under­stand where we came from and where we are going. We observe their move­ment to under­stand the beau­ty that encir­cles us every night as the sun sets. And we embrace its vast mag­nif­i­cence while iden­ti­fy­ing with our minute­ness in this tremen­dous­ly grand struc­ture that is the universe.

The writ­ten his­to­ry of astro­nom­i­cal obser­va­tion dates back over two thou­sand years. Observ­ing the night sky has evolved and devel­oped so that we now have an exten­sive list of space tele­scopes that bring us up close to the beau­ty of our cos­mos. It is utter­ly amaz­ing to think that a lit­tle over 2,000 years ago, this data-gath­er­ing process began to advance through the bril­liant work of the ancient astronomer Hipparchus.

Hip­parchus was born in 190 BCE in the King­dom of Bithy­nia, which today is now known as the region of North­ern Ana­to­lia, Turkey. We know about Hip­parchus through the writ­ings of ancient his­to­ri­ans, math­e­mati­cians, and sci­en­tists, includ­ing Ptole­my, who uti­lized Hipparchus’s astro­nom­i­cal find­ings for his infa­mous work Almagest. Ptole­my admired him so much that he referred to him as “that enthu­si­as­tic work­er and lover of truth.”[1] He has been ref­er­enced in the works of Stra­bo, who wrote Geog­ra­phy, and Pliny the Elder, who wrote Nat­ur­al His­to­ry. By the fourth cen­tu­ry, Hip­parchus had been ref­er­enced by Alexan­dri­an math­e­mati­cians Pap­pus, Theon, and Hypatia.

Image of a chord func­tion  and values

Unfor­tu­nate­ly, very lit­tle of Hipparchus’s works sur­vive. He wrote at least four­teen books, which includ­ed a star cat­a­log and one of the first trigono­met­ric tables in his work called Of Lines Inside a Cir­cle. This trigono­met­ric table includ­ed sev­er­al val­ues of a chord func­tion, which was quite a feat con­sid­er­ing his work came from 200 BCE.

Hip­parchus authored On The Length Of The Year, which were his obser­va­tions on the sun’s motions and orbits. He stud­ied the moon’s move­ment and deter­mined the peri­od of an eclipse by com­par­ing his data with Baby­lon­ian data from 300 years pri­or. When Hip­parchus flour­ished as a math­e­mati­cian and astronomer, it had already been known that the moon moved at vary­ing speeds. How­ev­er, no data showed the actu­al size of the orbits. Hip­parchus was the first astronomer to deter­mine the size of the moon’s orbit. Fur­ther­more, as not­ed by the great his­to­ri­an Pliny, the Elder, Hip­parchus was one of the first astronomers to show that lunar eclipses occur five months apart and that solar eclipses occur sev­en months apart.[2] He also revealed that the sun could be hid­den twice in thir­ty days, depend­ing on the viewer’s location.

Thus, many of our ear­ly astro­nom­i­cal find­ings would not have been real­ized if it were not for the works of Hip­parchus. His sys­tem­at­i­cal tech­niques helped him to dis­cov­er and mea­sure the Earth’s pre­ces­sion. And this dis­cov­ery was no Eure­ka moment. It was an exten­sive appli­ca­tion of trigonom­e­try, trigonom­e­try tables, appli­ca­tions of spher­i­cal trigonom­e­try, and geometry.

In physics, the Earth’s pre­ces­sion can be under­stood by look­ing at a gyro­scope or, more sim­ply, a spin­ning top. The top spins on its tip, and where one grabs it to make it spin is called the crown. The crown, as defined in physics, indi­cates the axis of the top. This axis is the line in which the body of the top spins around. When you spin a top, you will notice that the axis also rotates. As it speeds up, the axis makes a small­er cir­cu­lar motion. When it slows down, the axis makes a larg­er cir­cu­lar motion. The axis does not spin as fast as the top. The grav­i­ty of the Earth caus­es the axis to spin, which in physics is referred to as torque.

So, in the case of Earth, as the world spins on its axis while spin­ning around the sun, the axis, which is rep­re­sent­ed as the North Pole, is also spin­ning, albeit very slow­ly. This spin­ning is the Earth’s axi­al pre­ces­sion. His­tor­i­cal­ly, this was referred to as the pre­ces­sion of the equinoxes.

The Earth has two equinox­es, the spring equinox, and the fall equinox. An equinox is when the sun’s rays are per­pen­dic­u­lar to the Earth’s equa­tor. Thus, when the angle of the sun’s rays to the equa­tor is zero degrees.

Hip­parchus dis­cov­ered the Earth’s pre­ces­sion by fol­low­ing and mea­sur­ing the move­ments of the stars, specif­i­cal­ly Spi­ca and Reg­u­lus, two of the bright­est stars in our night sky. In his obser­va­tions, he mea­sured the lon­gi­tudes of these two stars. Then he com­pared his num­bers to the data of pre­vi­ous sci­en­tists and astronomers. He dis­cov­ered that the bright star, Spi­ca, had moved two degrees com­pared to its loca­tion dur­ing the fall equinox. Through this cal­cu­la­tion, he real­ized that the pro­ces­sion of our equinox­es moves at a rate between one and two degrees. Hip­parchus con­clud­ed that because the Earth’s axis moved so slow­ly, it would com­plete a rota­tion about every 36,000 years. This num­ber was fur­ther val­i­dat­ed in Ptolemy’s work Almagest.[3] What is impres­sive about this dis­cov­ery is that Hip­parchus was not that far off. We have since real­ized that the Earth’s axis com­pletes a rota­tion approx­i­mate­ly every 26,000 years.

In 398 CE, about 600 years after Hip­parchus, Syne­sius of Cyrene, one of Hypatia’s stu­dents and dis­ci­ples, wrote a let­ter to the mil­i­tary leader Pylaemenes. Along with that let­ter, Syne­sius had sent Pylaemenes an astro­labe. Synesius’s mis­sion was to influ­ence and encour­age the politi­cians of Rome to study and learn the val­ue of sci­ence. Clear­ly, the impor­tance of edu­cat­ing politi­cians on the val­ue of sci­ence has been an endeav­or among sci­en­tists for thou­sands of years.

In his let­ter, Syne­sius writes, “the great Ptole­my and the divine band of his suc­ces­sors were con­tent to have it as their one use­ful pos­ses­sion, for the six­teen stars made it suf­fi­cient for the night clock. Hip­parchus mere­ly trans­posed the stars and insert­ed them into the instru­ment.”[4] Thus, we have an ear­ly ref­er­ence that the astro­labe might have been Hipparchus’s invention.

Astro­labe

The astro­labe was the evo­lu­tion and com­bi­na­tion of an armil­lary sphere, a celes­tial map, and a diop­tra. An armil­lary sphere is a spher­i­cal frame of rings rep­re­sent­ing the stars’ celes­tial lat­i­tude and lon­gi­tude. It often has an axis that is char­ac­ter­ized by an arrow. If you have ever been shop­ping at Home­Goods, more than like­ly, you’ve seen a mul­ti­tude of them in the décor depart­ment, which is kind of cool.

A diop­tra was a mea­sur­ing tube with a pro­trac­tor. It sur­veyed over far dis­tances, which was use­ful for mea­sur­ing land for build­ing struc­tures and aque­ducts and for mea­sur­ing the posi­tion of the stars. Heron of Alexan­dria, in the first cen­tu­ry, ref­er­enced the diop­tra in his work The Diop­tra and indi­cat­ed that his instru­ment worked as a gen­er­al sight­ing tool and as a level. 

The astro­labe, a com­bi­na­tion of these three objects, allowed astronomers to map out the stars and project the night sky as a celes­tial sphere onto the plane of the equa­tor. The astro­labe even­tu­al­ly evolved into a flat, user-friend­ly, portable mech­a­nism. Metaphor­i­cal­ly speak­ing, with the astro­labe, users were then able to hold the galaxy in the palms of their hands.

The main body of the astro­labe is called the mater. The front part of the mater cra­dles the parts of the astro­labe togeth­er in the womb. At the top of the astro­labe is a cross, with twen­ty-four sym­bols etched around the limb with an M at the bot­tom. The cross rep­re­sents noon, and the M rep­re­sents mid­night. Etch­ings around the out­er rim rep­re­sent degrees, hours, or both.

The plates that sit inside the womb are called cli­mates. These cli­mates are mapped with a celes­tial sphere. The cli­mates can be inter­changed depend­ing on an individual’s lat­i­tude and loca­tion of obser­va­tion. All that the user can observe in three-dimen­sion­al space is flat­tened onto the plates of the astro­labe. Thus, the final tool that is need­ed to read an astro­labe is the imag­i­na­tion. So, if you were to hold an astro­labe in your hand, you would imag­ine the night sky as a dome of stars. This enor­mous, imag­ined sphere is a stere­o­graph­ic pro­jec­tion. Thus, stere­o­graph­ic pro­jec­tion is the map­ping of three-dimen­sion­al spher­i­cal images onto a two-dimen­sion­al plane, which, in this case, is the astrolabe.

Stere­o­graph­ic Projection

Stere­o­graph­ic pro­jec­tion is essen­tial for the astro­labe because it pre­serves cir­cles and angles. The astro­labe assists in deter­min­ing the angle at which one can see the moon or the stars. It also mea­sures alti­tude, lat­i­tude, and the width of rivers and val­leys. It serves as a com­pass and helps deter­mine the day’s hour.

How­ev­er, unlike a map that pro­vides pre­served dis­tances or areas on a ratio scale, stere­o­graph­ic pro­jec­tion cre­ates a pro­ject­ed map of curves ref­er­enced by inscribed angles.

So, when you hold the astro­labe, you are stand­ing on the Earth at a spe­cif­ic lat­i­tude. The lat­i­tude where you stand is the angu­lar mea­sure from Earth’s equa­tor. Look­ing up, imag­ine that you are stand­ing beneath of dome of stars.

I cre­at­ed the stere­o­graph­ic pro­jec­tion image with Hypa­tia stand­ing in Alexan­dria, Egypt, which is at 31° lat­i­tude. How­ev­er, she also stands at a 90° angle to her hori­zon. In the image, the green curve rep­re­sents the hori­zon. As Hypa­tia looks direct­ly up at the stars, her line of sight runs along the zenith and is per­pen­dic­u­lar to Hypatia’s hori­zon. The same line that runs below Hypa­tia and is per­pen­dic­u­lar to the hori­zon is the Nadir. In the image of Hypa­tia look­ing up at the stars, the entire area of the stars she observes is the almu­can­tar, also known as the observer’s lat­i­tude. When two stars lie on the same almu­can­tar, they have the same altitude.

The North Pole is per­pen­dic­u­lar to the equa­tor. Since Hypa­tia is stand­ing in Alexan­dria, Egypt, at 31° degrees lat­i­tude, the angu­lar mea­sure between her and the North Pole is 59°. As a result, the angu­lar mea­sure­ment from the South Pole to the Nadir is also 59°. The line that runs from the cen­ter of the almu­can­tar at Hypatia’s zenith to the South Pole inter­sects the equa­tor. This inter­sec­tion is the pro­jec­tion of almu­can­tars onto the flat plane of the astrolabe.

Thus, any observ­er look­ing at their astro­labe can ver­i­fy which stars they are look­ing up at by ref­er­enc­ing the almu­can­tars on the plates of the astrolabe.

Even today, the astro­labe serves as a valu­able tool for the astronomer. In addi­tion to observ­ing our heav­en­ly skies, trav­el­ers use this hand­held tool to nav­i­gate across land and water, sur­vey the height of build­ings or hill­sides, and esti­mate the lengths of rivers or oth­er distances.

The fol­low­ing image comes from a man­u­script titled Del modo di mis­urare, writ­ten by a 16th-cen­tu­ry poly­math Cosi­mo Bar­toli. Del modo di mis­urare con­sist­ed of sev­er­al books, includ­ing one ded­i­cat­ed to astrolabe’s mea­sur­ing capa­bil­i­ties. The con­cepts for deter­min­ing height and dis­tances from the back­side of the astro­labe use Hipparchus’s trigonom­e­try and trigono­met­ric ratios. 

Hip­parchus, thus, using this con­cept of stere­o­graph­ic pro­jec­tion, cre­at­ed a map by imag­in­ing a per­pen­dic­u­lar line that con­nect­ed each star to a point on the plates of the astro­labe. By using this astro­labe and observ­ing fixed stars, Hip­parchus was also able to mea­sure one’s geo­graph­i­cal lat­i­tude and the time of day or night at that geo­graph­i­cal lat­i­tude. And because he had such an exten­sive back­ground work­ing with trigonom­e­try and under­stand­ing the angles of pro­jec­tion, using a grade grid, he was able to assign a val­ue of lat­i­tude and lon­gi­tude to var­i­ous loca­tions on the Earth. These mul­ti­ple loca­tions of ref­er­ence allowed him to design the inter­change­able plates on the astro­labe that the view­er could change, depend­ing on where they were located.

This method of deter­min­ing the lat­i­tude and lon­gi­tude of geog­ra­phy con­tributed to his trea­tise Against The Geog­ra­phy Of Eratos­thenes. In this work, Hip­parchus rede­fined the car­tog­ra­phy of the world map by cor­rect­ing many of the geo­graph­i­cal mis­takes that Eratos­thenes made in his own work, Geog­ra­phy.[5]   

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At the Grif­fith Obser­va­to­ry in my home­town of Los Ange­les, there is a forty-foot mon­u­ment with a hol­low bronze armil­lary sphere at the top. There are six great astronomers carved into this mag­nif­i­cent mon­u­ment. The only one from antiq­ui­ty is Hip­parchus. And right­ly so. He was one of the very first astronomers who not only intrigued our curios­i­ty and imag­i­na­tion with stere­o­graph­ic pro­jec­tion but also defined the Earth’s geog­ra­phy. Addi­tion­al­ly, he was one of the first to not only observe but also math­e­mat­i­cal­ly and trigono­met­ri­cal­ly define his observations.

As we ride along with the stars and the galax­ies in our world, we dance among our own per­son­al con­structs of atoms, mol­e­cules, voids, and gath­er­ings. Hipparchus’s math­e­mat­i­cal astron­o­my ground­ed us in under­stand­ing where we are in the world and the uni­verse. He helped us to see the chore­og­ra­phy of the uni­verse and showed how we move with it. Thus, his obser­va­tions piqued our curios­i­ty and inspired us to imag­ine our place as we stand on this lit­tle blue dot mov­ing through space as observers and par­tic­i­pants in this glo­ri­ous dance of the stars.

Until next time, carpe diem!

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[1] Ptole­my, 2nd cent. “The Almagest.” edit­ed by R.M. Hutchins, Vol. 16. Great Books of the West­ern World. Chica­go: Chica­go : Ency­clopae­dia Bri­tan­ni­ca, 1952. http://archive.org/details/almagest00ptol: ix.

[2] Pliny, the Elder. The Nat­ur­al His­to­ry of Pliny (Vol­ume 1). Edit­ed by John Bostock and H.T. Riley. Vol. 1. Lon­don : Hen­ry G. Bohn, 1855. http://archive.org/details/57011150RX1.nlm.nih.gov.

 

[3] Gold­stein, Bernard R., and Alan C. Bowen. “The Intro­duc­tion of Dat­ed Obser­va­tions and Pre­cise Mea­sure­ment in Greek Astron­o­my.” Archive for His­to­ry of Exact Sci­ences 43, no. 2 (1991): 93–132.

[4]. Augus­tine Fitzger­ald, “On an Astro­labe,” in The Let­ters of Syne­sius of Cyrene. Trans­lat­ed into Eng­lish with Intro­duc­tion and Notes (Lon­don: Mil­ford, 1926), 263.

[5] Shche­glov, Dmit­ry A. “Eratos­thenes’ Con­tri­bu­tion to Ptolemy’s Map of the World.” IMAGO MUNDI-THE INTERNATIONAL JOURNAL FOR THE HISTORY OF CARTOGRAPHY 69, no. 2 (Jan­u­ary 1, 2017): 159–75. https://doi.org/10.1080/03085694.2017.1312112.