Number Theory in the Seventeenth Century
Number theory, also earlier on known as arithmetic, primarily refers to an expansive and intriguing domain of theoretical and conceptual mathematics (Stillwell, 2011). Stillwell (2011) asserts that number theory also occasionally goes by the name prime calculus since it entails the analysis of numbers written without fractional components. Therefore, mathematicians specializing in number theory mainly base their study in prime numbers including probes, evaluations and surveys related to items acquired from integers such as rational numerals. Furthermore, the said specialists also usually use various scientific and methodical objects to answer questions occurring in forms of number theory. In essence, these logical items tend to include several functions such as the Riemann zeta, totient and divisor functions among others which aid in the systematic and fashionable encryption of features and attributes of either whole, prime or other items associated with number theory (Ore, 2012). As such, this essay brings out various statistical, geometric and algebraic ideas, concepts, and facts as per the number theory during the seventeenth century by mainly focusing on several mathematicians during the said era such as John Napier, Sir Isaac Newton, Blaise Pascal, Pierre de Fermat, Rene Descartes, and Leibniz.
Undoubtedly, the seventeenth century saw a revolutionary and exceptional eruption of both arithmetical and research based notions and concepts throughout the entire Europe following the renascence and revival period. Essentially, this phase is usually sporadically known as the epoch of cogitation given that many specialists made unprecedented discoveries during this era (Ahlfors, 1953). First, the creation of logarithms by a mathematical expert known as John Napier during the primitive phases of the seventeenth century enhanced the improvement and progression of several sectors such as mathematics, study of the stars and planets apart from earth and physical sciences. The advancement was achieved since Napier’s invention of logarithms made previously seemingly strenuous and grueling calculations reasonably uncomplicated and elementary.
Napier generated that the logarithm of a numeral tends to be the proponent after subjecting the integer to either ten times multiplication or any other pedestal (Ahlfors, 1953). In addition, the final results of the computation of logarithms surfaces from the actuality that the augmentation of two or more integers tends to correspond with the addition of their logarithms. Similarly, attaining the value of the division of either two or more numbers entails the deduction of their logarithms (Ahlfors, 1953). In the same way, obtaining the square values of given numerals simply involves compounding the logarithm by two or in cases that require cubing, the logarithm should be multiplied by three. On the contrary, in order to attain square root values of specified numbers, this implies the division of the identified logarithm by two and by three in order to acquire cube root (Ahlfors, 1953).
Moreover, Napier also postulated that even though base ten happens to be the most sought after while computing mathematical and scientific problems, there exists another collective base by the number e . Apart from possessing the worth of 2.7182818, the said base also contains a number of remarkable, unique and outstanding attributes and features that in turn render it extremely trivial for mathematical computations on the total logarithms (Dickson, 2005). Besides, Napier also ameliorated the whole concept of the decimal, originally generated by a specialist known as Simon Stevin, by universalizing the applications and utilities of the decimal point. In the same way, Napier also made Al-Khwarizmi’s notation of grid multiplication extra favorable and suitable by introducing a multiplication device that he named Napier’s bones (Dickson, 2005).
Similarly, another prominent and highly esteemed chief character in the seventeenth century’s empirical metamorphosis was a man from France known as Rene Descartes. In fact, Descartes is still up to date occasionally reckoned as the leading individual in the contemporary department of mathematics (Dickson, 2005). In addition, Dickson (2005) claims that other people have also unanimously rendered Rene as the sole inventor and father of the present day philosophy. Nevertheless, Descartes did not entirely begin as a mathematician or any other specialist of that kind. Instead, before he later on discovered and acknowledged his passion in seeking real intelligence and scientific knowledge, Rene had secured a full time job as a mercenary soldier during his earlier days. Dickson (2005) notes that after experiencing and undergoing a succession of aspirations, inspirations and yearnings as well as his contact with Isaac Beckman, who happened to be an influential theorist and researcher, Descartes finally turned around. Therefore, instead of continuing with his first career as a soldier, Rene ventured into a new field of science where he majored in both mathematics and in the study of energy, force and power.
Here, the unsophisticated Descartes shortly discovered that to understand and become successful in the extremely ambiguous, unpredictable and uncertain domain of philosophy, the only way out was to found the subject on the conclusive and patent certainties and figures of mathematics (Kilpatrick, 2014). However, in order to accomplish his newly found dream, Descartes had to relocate to Netherlands whose surroundings gave room for more freedom of thoughts unlike the extremely restrictive catholic based France. Once in Netherlands, Rene commenced working on his relentless dream of achieving his goals and objectives of amalgamating the mathematics of universalized arithmetical operations with the broad sector of the study of superficial appearances of things as picked out from the main substance (Kilpatrick, 2014). As such, after engaging in sufficient and viable research towards the attainment of his dream, Descartes finally promulgated and publicized his metaphysical and arithmetical exposition and dissertation in sixteen thirty seven, which turned out to be both radical and revolutionary. Besides, one of the treatise’ supplements which Rene named “la Geometrie” currently appears as a monument in the life story of the invention of mathematics.
Furthermore, Kilpatrick (2014) claims that it was through the monumental epilogue that Descartes first brought out his ideology on Cartesian coordinates. Essentially, these set of coordinates suggest that two integers on a plane can represent each detail or element in two given measurements. This is so because, while one of the numbers describes the element’s horizontal locale, the other one indicates its plumb locus. Therefore, to represent the said idea, Rene used upright or vertical lines traversing at a specific point termed as the origin in order to quantify both the parallel and perpendicular positions also referred to as the x and y respectively. Moreover, the said locations tend to appear in both positive and negative sides thereby virtually partitioning a flat surface into four equal parts (Kilpatrick, 2014). Thus, by plotting the answer set of an equation on the plane, one can easily illustrate any given equation. For instance, going by an elementary mathematical problem that point y equals point x, it results to the formation of a straight line connecting several coordinates such as (0,0), (2,2) and (4, 4) and others appearing on the said line (Kilpatrick, 2014).
Here, the chief point of focus is that any equation represented on a plane should always yield a straight line that in turn allows the materialization of the required points according to the equation. Nevertheless, more sophisticated equations comprising squares and cubes produce several types of crescents on the given horizontal surface (Ore, 2012). Furthermore, the coordinates of a point change with its movement along a curve thereby meaning that different positions marked on a curve would portray varied identified positions corresponding to an axis. However, the said ideology is coherent in that the denoted change can be delineated using a written equation regardless of the position of coordinates on the plane (Ore, 2012). Therefore, going by this novice and innovative technique, equations depicting the summation of the squares of both x and y to be equal to four could also be effective in drafting graphs of quadratic functions. Hence, in the given scenario, the resulting parabola of the said equation would be deducting sixteen units of x from the square of y.
Moreover, the said revolutionary mathematical and philosophical creation known as analytic configuration permits the reversible transformation of the mathematical parts whereby both formulae and equations use letters as well as other general signs to symbolize different integers and quantities, into geometry. Thus, this means that due to the unique ability of Cartesian geometry, solving equations that occur simultaneously can be achieved through either using the graphical or algebraic strategies (Stillwell, 2011). In turn, this invention not only simplified working out simultaneous equations but also acted as a foundation for other Descartes’ close contemporaries referred to as Newton and Leibniz in their succeeding pioneering of calculus. In the same measure, the Cartesian geometry also completely transformed mathematics through the generation of concepts extremely important and relative to the present subject dealing with the art of science as well as the study of interactions between matter and energy (Stillwell, 2011). Here, the modern technology using Descartes’ geometry brings out itself in the sense that graphical plotting of orbiting of planets is currently possible.
In addition, another highly esteemed individual who implicitly generated the up to date number theory was known as Pierre de Fermat (Kilpatrick, 2014). The said man practically crafted the number theory solely regardless of the fact that he was both incompetent and unskilled while starting the venture. Pierre acquired his inspirations to pursue the mathematical discipline from Diophantus, a renowned mathematician who invented the “Arithmetica” during the Hellenistic period. Because of Diophantus’ stimulation, Fermat devised and contrived a number of pristine integer models and prototypes, which had since time immemorial proved difficult to many skilled mathematicians. Besides, Ahlfors (1953) asserts that Pierre also formulated a series of postulations, inferences, axioms, laws and principles throughout the course of his life. More so, Fermat also takes the acknowledgement and glory associated with his timely evolution of that fore shadowed the contemporary calculus as well as theories associated with probability. However, most of Pierre’s mathematical works never got published since he instead opted to send letters and marginal notes to his friends and fellow counterparts (Ahlfors, 1953). Some of Fermat’s propositions deduced from general postulates include the Two Square Theorem which posits that any given archetypal digits or units that sums up to one when duly divided by four automatically expresses itself as four to the power of n with the addition of one.
Additionally, another symbolic representation related to Pierre was one which he termed as the Little Theorem. Even though he named it little theorem, this supposition is extremely vital and helpful mostly in the modern world, since experts use it to safeguard and preserve credit cards in world wide web business deals and undertakings (Stillwell, 2011). This theorem in numerous occasions applies in the inspection and investigation of sizable and substantial prime digits. In other words, the Little Theorem implies that given two units namely b and c, whereby c is a prime integer but does not occur as a factor of b, then by multiplying b by itself equivalent to c less one times and ultimately divide the answer with c, one will always get the remainder of one (Stillwell, 2011). In the same way but in a different measure, Pierre also established a subcategory of units, which he named the Fermat integers. Intriguingly, all the subsets tend to be prime digits commonly known as the Fermat primes and they comprise the addition of one to a number less than two to the double power of two.
Nevertheless, other specialists and mathematicians have made conclusions that the other identified advanced Fermat integers go against the prime number rule, which in turn shows the need for prefatory and preliminary verification and authentication in the sensitive discipline of mathematics. As a result, such validation and confirmation goes a long way in ensuring the disposition of correct mathematical and metaphysical facts, figures and theorems (Ore, 2012). Ultimately, Pierre’s final mathematical work was one he called “piece de resistance” which he sadly left unverified during the time of his demise. This particular theorem gave mathematicians sleepless nights and troubled days for approximately three hundred and fifty years (Stillwell, 2011). Fermat relayed this supposition using a hurriedly written note on the edge of his replica of his role model’s, Diophantus, creation.
Here, Pierre claimed that three positive numerals say x, y and z can at no given point satiate or gratify the mathematical problem denoted by x to the power of n with the addition to y to the power of n will finally accrue to z to the power of n provided that n is equal or greater than two (Dickson, 2005). Nevertheless, Pierre interestingly asserted to possess viable substantiation of the said difficult theorem but instead chose not to let it out by claiming that the brink did not leave enough space. Therefore, Fermat as well as all the other mathematicians who tried to decipher and authenticate the “piece de resistance” theorem only partially achieved to verify the exceptional case by taking into consideration that n equals to four. As a result of the level of difficulty posed by the unsolved theorem various academies offering both arithmetical and scientific disciplines volunteered considerable and valuable prizes to no avail.
Nonetheless, efforts put in by different specialists and mathematicians to crack the theorem’s code in a way pioneered the evolution of the number theory of algebra during the nineteenth and twentieth centuries. Furthermore, Pierre also to some magnitude foresaw the invention of calculus during his investigations on the suitable methods for obtaining and locating the cores of gravity for different flat surfaces and solid motifs (Dickson, 2005). Here, Fermat innovated simpler techniques of attaining the upper limits, minima and digressions to a number of arcs and the invented strategies proved to be almost identical to the method of separation. Later on, Pierre’s innovations turned out to be immensely indispensable and helpful to other scientists such as Newton and Leibniz. Moreover, Pierre’s commensuration with one of his earlier counterparts known as Pascal also undoubtedly aided specialists and researchers to comprehend an essential notion in fundamental probability.
Likewise, Blaise Pascal was also a leading and influential researcher, theoretician and algebraist during the seventeenth century (Kilpatrick, 2014). Just like many specialists and mathematicians, Pascal mastered many skills and arts at a tender age and as such, he followed a number of pursuits linked to analytical and knowledgeable activities, schemes and projects in the course of his life. However, during his early endeavors, Pascal concentrated more in logical and practical sciences where he came up with a law of physics named after him. This law states that the amount of tension applied at any point in an incarcerated solution tends to be evenly spread or disseminated throughout the entire solution. In the same manner, Pascal was also an expert in mathematics and at the primal age of sixteen, he invented a remarkable exposition on the discipline of extrusive geometry that is commonly referred to as Pascal’s Theorem (Kilpatrick, 2014). In essence, this disquisition posits that in the case where a hexagon becomes incised in a circle, the resulting line from the three opposite areas of intersection is usually termed as the Pascal line. Besides, while still at his early stages of life, Pascal also invented a running and operative device for performing calculations based on elementary subtractions and additions that aided his father in computing tax problems.
However, Pascal is relatively famous for his invention of the Pascal triangle which acts as an appropriate and favorable technique of plan display of bivalent coefficients whereby each presented integer tends to be the sum of the two digits immediately overhead (Stillwell, 2011). Therefore, in cases where binomial equations become expanded, it is mandatory for the resulting coefficients to form an evenly shaped triangle. Even though Blaise was not the first mathematician to come up with the binomial triangle, he submitted a graceful and sophisticated authentication that revolved around defining the numerals via certain rules and formulae (Stillwell, 2011). Furthermore, Pascal also came up with a number of intriguing and productive designs in between the lines, successions and processions of the assortment and collection of numerals.
Moreover, Stillwell (2011) claims that Blaise also generated the theoretical ideology of using the Pascal triangle in various ways such as in aiding the computation of mathematical problems related to probability. As a matter of matter, the probability theory evolved from the combined efforts put in by Pierre Fermat, Christian Huygens and Pascal Blaise. Through this correspondence, the mathematicians and scientists invented the idea of evenly probable end results whereby the likelihood of an event or phenomenon would be calculated through computing the total number of expectable aftermath of the entire situation (Stillwell, 2011). In turn, this not only simplified the exercise but also permitted the use of both ratios and fractions in the computation of possible outcomes via summations and multiplications.
Finally, in the exhilarating ambience characteristic of the seventeenth century, resulting from the enlargement of the British territory, most universities and institutions of higher learning ended up yielding several prominent and important researchers and mathematicians (Ahlfors, 1953). Nonetheless, the most distinguished and acclaimed scientist, astronomer and natural theorist indeed as Sir Isaac Newton. As such, many people globally refer Newton as the most powerful and dominant human being over the years due to his many scientific innovations and inventions through broad based research and experiments such as the apple tree and gravity (Ore, 2012). Here, the invention of calculus allowed architects, developers and number theorists to make sense of the relentless shifting, magnetic and forceful alterations of the normal world which transforms and transmutes daily, unlike the fixed framework discovered by the Greeks.
Moreover, Newton also aspired to represent slopes of curves and arcs that tend to vary from time to time (Dickson, 2005). Hence, he implored Gottfried Leibniz a fellow scientist and mathematician and together they created an imitative and obtained function whose main purpose was to represent all points of a curve. Newton’s obtained function f into x or f (x) is commonly termed as either divergent calculus or the technique of fluxions according to Newton. Following the same ideology, Newton created the other rule of integration which functions in a directly reverse way from differentiation and the two methods comprise the chief methodologies of calculus (Dickson, 2005). Finally, people also acknowledge and credit Newton for the creation of the binomial dissertation. The said theorem systematically outlines and delineates the increment in size of the powers of a binomial using algebra.
Indeed, the history of the number theory is both fascinating and immensely intriguing as shown throughout the essay. The outlined mathematicians, alchemists, scientists, and philosophers had their contributions to the invention of mathematics and various scientific theorems, which still exist up to date as clearly brought out in the paper. For instance, not only did Napier’s invention of logarithms act as a measure of success for him, but it also aided in simplifying mathematical problems, which previously proved hectic. Similarly, even though he was not the original creator, Napier also revolutionized Simon Stevin’s ideology of decimals. In the same way but in a different measure, even though Descartes did not start out as a mathematician, it was via him that Cartesian coordinates, which helped in solving simultaneous equations came to be. Here, through Isaac Beckman’s continued support, Descartes was able to put himself together and become unstoppable in the domain of mathematics and philosophy whereby he experienced the materialization of his dreams. Likewise, Pierre de Fermat through the application of Diophantus’ works was able to crack and simplify an extremely previously difficult mathematical concept of number of pristine numeral models and prototypes. This achievement together with the many others he came to possess through the course of his life also chipped in greatly in the invention of the number theory. Additionally, the other theoretician and mathematician named Blaise Pascal also contributed in the said invention by mainly having inputs on what he called the Pascal triangle. Besides, as shown in the essay, it was through him that the theory of probability finally came to be after being postulated and discussed earlier on by other specialists. Ultimately, Newton finalized the invention of the number theory in the seventeenth century through the promulgation of his ideas on gravity as well as his long term desire to represent the varying slopes of both curves and arcs.
References
Ahlfors, L. V. (1953). Complex analysis: An introduction to the theory of analytic functions of one complex variable. New York: McGraw-Hill.
Dickson, L. E. (2005). History of the theory of numbers. Mineola: Dover Publications.
Kilpatrick, J. (2014). History of research in mathematics education. InEncyclopedia of mathematics education (pp. 267-272). Springer Netherlands.
Ore, O. (2012). Number theory and its history. Mineila: Dover Publications.
Stillwell, J. (2011). Mathematics and its history. New York: Springer Publications.
