The Mathematics of The Quadrivium

by Brian K. Davis

Arithmetic, geometry, astronomy, and music are the four subjects that make up the quadrivium. A term first coined in the medieval period, it still carries weight today. Although it would appear that the quadrivium is made of two mathematical subjects and two non-mathematical subjects, astronomy and music are also based in math. Thus the quadrivium was solely based in the art of mathematics. Math has evolved over time and is still evolving today. The first known study of math came from Mesopotamia in 3,000 B.C.E. with the Babylonians and the Egyptians (Kline 3). However this math was basic and relied mostly upon counting. Math would not grow as a subject until the time of the Greeks in about 775 B.C.E. but would be lost to the world until the fifth century when Anicius Manlius Severinus Boethius (c480-524) began translating the Greek works (Kline 201). After Boethius’ translations of Greek math into Latin do we see the rise of math in Europe to help explain the universe. The use of the quadrivium would give scholars the mathematical ability to analyze the physical world around them, the first of these scholars being the Greeks and Plato’s Republic.

Plato’s book the Republic was centered on creating a utopian society for the people of Greece. Plato would also go on to describe a perfect school system to teach those who would eventually lead society. In his definition of education, comes the first inkling of the quadrivium, although not outright stated. When talking about education, Plato believes that the science of arithmetic is wholly concerned with numbers and the quality of numbers leads to truth (Plato 161).

Plato believes that by understanding numbers, people can better understand the truth and gain insight into the world. Plato would also go on to state that numbers are important for generals as well as philosopher, “For a soldier must learn them (arithmetic) in order to marshal his troops, and a philosopher, because he must rise out of the region of generation and lay hold on essence or he can never become a true reckoner, (Plato 161)” Plato believes that understanding arithmetic will better organize the troops in battle. However the nuance of truth keeps cropping up in Plato’s statements as well, “This branch of learning (arithmetic) should be prescribe by our law and that we should induce those who are to share the highest functions of state to enter upon that study of calculation and take hold of it, not amateurs, but to follow it up until they attain to the contemplation of the nature of number, by pure thought, not for the purpose of buying and selling, as if they were preparing to be merchants or hucksters, but for the uses of war and for facilitating the conversion of the soul itself from the world of generation to essence and truth, (Plato 163).” Plato likes the study of arithmetic because it is tangible, meaning it can be easily proven or disproven making people think about their arguments before making one (Plato 165).

After Plato’s discussion on arithmetic comes geometry. Once again he connects geometry to war, “So much of it (geometry),” he said, “as applies to the conduct of war is obviously suitable. For in dealing with encampments and the occupation of strong places and the bringing of troops into column and line and all the other formations of an army in actual battle and on the march, an officer who had studied geometry would be if he had not,” (Plato 167) and also stating, “Its (geometry) uses in war, and also we are aware that for the better reception of all studies there will be an immeasurable difference between the student who has been imbued with geometry and the one who has not,” (Plato 173). Plato also mentions how geometry compels the soul to contemplate essence, and that geometry forces the soul to turn its vision round to the region where dwells the most blessed part of reality (Plato 169), Plato believed that geometry would help people examine the physical shapes around them thus opening their eyes to truth, “For geometry is the knowledge of the eternally existent. Then, my good friend, it would tend to draw the soul to truth, and would be productive of a philosophic attitude of mind, directing upward the faculties that now wrongly are turned earthward,” (Plato 171).

After astronomy would follow geometry in which Plato was quick to point out its uses for telling the seasons, months, and navigation (Plato 171), all of these being important for the study of war. Astronomy also makes the soul look upward and leads it away from things here to those higher (Plato 181). Plato also believed that the study of the stars would lead towards reality, “We must use the blazonry of the Heavens as patterns to aid in the study of those realities, just as one would do who chanced upon diagrams drawn with special care and elaboration by Daedalus or some other craftsman or painter,” (Plato 185). Plato did not have much to say on music other than astronomy was for the eyes and harmonies for the ear (Plato 189). Instead he refers to the Pythagorean’s writings about music, “They (Pythagoreans) transfer it to hearing and measure audible concords and sounds against one another,” (Plato 191). Pythagoras was the first person to examine the physical science of music. Kline would state this about Pythagoreans, “Because the Pythagoreans “reduced” astronomy and music to number, these subjects came to be linked to arithmetic and geometry; these four were regarded as the mathematical subjects,” (Kline 149).

Long before anything was known of pitch numbers, or the means of counting them, Pythagoras had discovered that if a string be divided into two parts by a bridge, in such a way as to give two consonant musical tones when struck, the lengths of these parts must be in the ratio of these whole numbers. If the bridge is so placed that 2/3 of the string lie to the right, and 1/3 on the left, so that the two lengths are in the ratio of 2:1, they produce the interval of an octave, the greater length giving the deeper tone. Placing the bridge so that 3/5 of the string lie on the right and 2/5 on the left, the ratio of the two lengths is 3:2, and the interval is a fifth,” (Helmholtz 14). Pythagoras would later build a tuning scale based on fifths which was used until the creation of just intonation and tempered tuning. Plato would later state astronomy and music as a useless form of numbers; however this section of the Republic did set the stage for later works on the quadrivium.

Boethius was a medieval scholar that served a great purpose in the expansion of the quadrivium. Boethius would translate some of the Greek works on math and would create the term quadrivium in his own treatise on math. Boethius would get his idea for De Arithmetica from the Greek thinker’s liberal arts curriculum (Masi 83). Boethius would re-introduce the Greek works to the western part of Europe. In his book De Arithmetica Boethius would introduce the idea of proportionality, “He begins his discussion of proportionality with an extensive list of the types, drawn directly from the Boethian De Arithmetica. A ratio is a relation between two terms, as 1:2, or as expressed in a fraction, ½ and the relationship is called a rational number in its fractional form. A proportion is a ratio between ratios, as when 1 compared to 2, which is as 2 is to 4. Proportion may be set up in series, as a series of duplex, triple, or quadruple proportions. Bradwardine extracts the idea of proportionality from the Boethian conception of proportion. Bradwardine adapted, also from Boethius, the idea that the most important proportionalities are the arithmetic, geometric, and harmonic,” (Masi 91).

Boethius would also introduce irrational numbers as a way to explain geometry, like the sides of the triangles (Masi 91). Outside of translating the Greek works, Boethius’ works would become popular in the Middle Ages, “We must conclude that the Boethian mathematics enjoyed an extraordinary increase in popularity and influence between 1200 and 1600,” (Masi 81). His book would even become the text book during the middle ages, “Moreover, for the teaching of the first of the quadrivial arts, arithmetic, the Boethian De Institutione Arithmetica appears to have maintained its position as a basic text, and his was the case despite the fact that there were available for the study of arithmetic in the thirteenth century, in its various practical as well as other aspects, a wealth of materials both old and new,” (Kibre 72). Boethius’ works would go on unchallenged until Roger Bacon. However these two both believed that education needed a solid base of mathematics (Masi 92). Bacon would disagree with the order in which the quadrivium be placed, however he did agree with Boethius on the teaching of arithmetic, the species of numbers and the reasons for their operations (Kibre76). Boethius would also write a work on music.

Music was another sub-category of the quadrivium and was another topic Boethius would translate Greek works and write his own treatise on. During the Medieval and Middle Ages, chant was the source of music. Chant was only used by the Catholic Church and was in Latin, the chants were used to help the common people memorize scripture. The irony of this last statement is that most people could not speak Latin thus they did not know what they saying. The first part of Boethius’ book dealt with chant and how it should be organized. Boethius would organize the chants into tonaries, or the classifying of chants according to their tonal and melodic similarities (Bower 164). Boethius would place the chants based off their church modes which were: Lydian, Dorian, Myxolydian, Phrygian, Hypo-Phrygian, Hypo-Dorian, Hypo-Lydian, and Hypo-Myxolydian, church modes are based off which note they start on. After establishing the tonaries, Boethius would move on to compiling works into two categories, practical tonaries, those used by professional singers to check the tone of a chant, and didactic tonaries, those used to teach students the basic qualities of each tone (Bower 164). Boethius would treat music differently compared to his contemporaries. He took a more analytical approach whereas his peers believed in the mysticism of music. To Boethius, the main approach to musical pitch was qualitative rather than quantitative (Bower 165).

Boethius believed that one could mathematically analyze music in order to learn how to better write music; in essence he created the first step towards music theory. Boethius affirms that one holds immutable truths concerning music when one knows the related mathematical quantity in the proportions of consonances (Bower 166), in Boethius’ time this would be the examination of one tone to another. For example when one strikes a string and touches it in the middle the octave is sounded thus the relationship between a note and an octave must be 1:2. Boethius would base his math off of Pythagoras. Boethius also argues that the practicing artist is separated from musical science, and thus is not worthy to be named a musician (Bower 166), in other words if a musician does not understand the theory behind it they do not fully grasp it and in fact are missing an element to music. Boethius’ works would begin the study of music theory, however like his other works, failed to advance the study of math as a whole.

The problem with the quadrivium was that it was translated into Latin but never expanded upon. In the Middle Ages the quadrivium was arithmetic, considered as the science of pure numbers; music, regarded as an application of numbers; geometry, or the study of magnitudes such as length, area and volumes at rest; and astronomy, the study of magnitudes in motion (Kline 202), and did not really evolve past that definition. According to Kline, “the introduction of some of the Greek words retarded the awakening of Europe for a couple of centuries. By 1200 or so the extensive writings of Aristotle became reasonably well known. The European intellectuals were pleased and impressed by his vast store of facts, his acute distinctions, his cogent arguments, and his logical arrangement of knowledge,” (Kline 207).However the quadrivium would have an effect on the Church. Once established, the clergy was expected to defend and explain the theology and rebut arguments by reasoning, and mathematics (Kline 202). Today the quadrivium is still used; however it is not referred to as such.

Math is now taught to every child in the United States. Most kids will start their training in mathematics in elementary school. Arithmetic is still the first subject taught with addition and subtraction, followed by the basics of geometry. What is interesting is that education today combines basic arithmetic and geometry with multiplication. Astronomy is rarely taught and the basic of music, mainly how to play an instrument, is taught. If one is lucky they will receive a little music theory, however most students do not. Education today however, has grown beyond the quadrivium. Today students are taught trigonometry, an upper level math of geometry, and advanced algebra, upper level arithmetic, and calculus. With the advancement of technology, there is less need for the average person to know astronomy, however for those who go on to study astronomy as their profession will rely on calculus. The study of music has advanced beyond comparison of tones.

Scientists have figured out that music is a disturbance of air in a wave like pattern. Hermann Helmholtz would crack many mysteries behind musical sound. Helmholtz was able to prove that musical sounds were actually complex sine waves, “Where two condensations are added we obtain increased condensation, where two rarefactions are added we have increased rarefaction; while a concurrence of condensation and rarefaction mutually, in whole or in part, destroy or neutralize each other,” (Helmholtz 28), in other words, sounds are created by multiple sine waves that are added together. Helmholtz proved there were multiple sine waves with his invention the Helmholtz resonator which is a bottle that will resonate only one sound, one sine wave, which is part of the complex sine wave (Helmholtz 43). However, this is a more advanced analysis of music. Most students will not learn this unless they go into music as a profession, even then they may not receive this knowledge. Professional musicians, however, will get a healthy dose of music theory. Students today no longer learn only the quadrivium; in fact, students now learn more math and its applications than at any other point in history.

The quadrivium was first discussed by Plato in what he believed was necessary to be a philosophical person. Boethius would later translate the Greek works of Plato, Aristotle, Euclid, and many others into Latin and introduce the quadrivium to the school system of Western Europe. Boethius’ quadrivium would be the platform for which the study of math would stand on for centuries and would later become the first step in expanding the concepts of math. Today the quadrivium is no longer directly taught but its subject matter still is. Students get a great deal of arithmetic and geometry and some will focus their studies on astronomy and music. The quadrivium served an important purpose in the advancement of European education and its affects should not be overlooked.

Bibliography

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This paper was originally created for Steve Jackson’s History of Higher Education course.

 

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