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The science which bears this name is a generalised extension of the science of number. From this consideration Sir Isaac Newton was led to give to it the title of “Universal Arithmetic.” This designation suggests a more correct idea of its nature than the inadequate name handed down by tradition. Besides, it appears calculated to lead the student to such a knowledge of the general operations of the science, as to render his comprehension of ordinary mathematical language more intellectual and less mechanical. As this science depends primarily on the same fundamental principles as Arithmetic, this necessary connection is required to be kept constantly in view. The following brief notices will be restricted to a general sketch of the origin of the science, as far as it has been ascertained, and the progressive improvements and discoveries. These will be seen to have been very slow and gradual in different countries and in different ages ; and when viewed in succession, to extend over centuries.

The history of the first steps in the development of numerical systems, as of written languages, is involved in uncertainty, and it is perhaps impossible to ascertain the origin of any of the systems which existed among the people of early times. The methods of arithmetical notation employed by the Hebrews, the Phænicians, and the Egyptians, and the still ruder systems employed by the Romans, are all of them limited and imperfect, and the remaining monuments of their literature and antiquities afford no intimation whatever that they were acquainted with any superior system. The powers of numeration at first must have been very limited before the art of writing was invented. A comparison of the written names of numbers found in the records of ancient nations may lead to some probable conjectures respecting their possible connection in more early times. However interesting it might be to trace the steps by which mankind originally applied and gradually extended their first essays of number to the various occasions of human life, there are no extant writings, known at present, which afford the needful assistance.

The art of denoting and naming numbers, in some form, became necessary among men as soon

as the distinction of property was acknowledged, which appears to have been coeval with the origin of society. The earliest methods of numbering would naturally be limited to the naming of such numbers as the necessities of the people required.


The earliest extensions of the primitive mode would be made by the people whose necessities extended beyond the wants of a patriarchal and pastoral mode of life. In the early ayes of which any records exist, some progress had been made in the arts of number and measure, as is clear from the narrative of the life of Abraham in the twenty-third chapter of the book of Genesis. The next improvements would be found in those nations who applied themselves to commerce and navigation. It is therefore among the earliest civilised nations which directed attention to commerce, that arithmetical calculation could have received its greatest improvement. It is recorded of the Egyptians and Phænicians, that these two nations made the first improvements in the practice and calculation of numbers. The necessity of improved methods would be found in the management of the public revenues of states and kingdoms, and men would be naturally led to find out methods of abridging and improving the operations in which they were every day engaged. The account of the revenue in Egypt when Joseph was the chief minister, implies that their system of arithmetic was not of a very complete nature for the expression of large numbers; as it is recorded in the forty-first chapter of the book of Genesis, that Joseph left off numbering the quantity of corn laid up in the cities, “ for the quantity was without number;" an expression which can only refer to the system of numbering known in Egypt at that time.

Previous to that period, their ordinary methods might have been sufficient for the necessities of the government in the regulation of the

At a later period, it was in Egypt that Pythagoras learned the theories concerning the nature and properties of numbers, which he taught in the school he founded at Crotona, in Magna Græcia.

The Phænicians, in the remotest ages, applied themselves to commerce and navigation, which required them to be conversant with numerical calculations. Ancient historyl ascribes to them the invention of casting accounts, and keeping registers of everything that relates to the affairs of merchants. Plato? writes that a sophist affirms of the Lacedemonians, that they scarcely knew how to number. This censure of the ignorance of the people of Lacedæmon, seems to imply that they did not frequent the school in the grove of Academus at Athens, where that science, with others, was sedulously taught and cultivated.

By reducing the elementary processes of numbers to their first principles, somo attempt may be made to discover those operations which, on account of their simplicity, might naturally present themselves to the mind. The operations of Arithmetic depend on the two simple


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1 Strabo, B. xvii., p. 1136, B.
* Hippias Major, p. 1248, A. fol. Francofurti, 1602.


processes of addition and subtraction. Multiplication of integers is nothing more than the addition of equal numbers. Division bears the same relation to subtraction as multiplication does to addition. It is therefore necessary that reference should be made to addition and subtraction for the origin of the methods of calculation. Addition and subtraction presuppose numeration, the source which furnishes Arithmetic with the material of all its operations. To count is nothing else than to form ideas of different sets or assemblages of counters, and to give names to each of the sets or groups. This forms the first act of the mind in learning the science of numbers. Every particular object brought before the sight suggests the idea of one thing or unity to the mind, and every assemblage of objects or units suggests the idea of numbers, or of a greater or a less assemblage of units. The idea of a simple number is an abstraction, and from the fact of men having five fingers on each hand, they would naturally be led to form ideas of as many sets of other objects as are equal to the number of fingers on one or on both bands. The fundamental notions of number most probably arose from the use of the fingers on one or both hands as an instrument of calculation. Names to express these elementary numbers would be devised, perhaps derived from the names of the instrument itself used in counting. Arithmetic most probably began with practical numeration, or determining the number of several objects, as the number of a herd of cattle or a flock of sheep. As men are provided with a kind of calculating instrument, it is highly probable that the fingers of one or both hands were the first instruments used to assist them in the counting by sets. Homer represents Proteus' as counting his sea calves by fives—that is, by his fingers—and employs the word neuráželv, which literally means to count by the five fingers; so also Æschylus uses the same word? for counting out from the urn the black and white pebbles which expressed the verdict of the judges after the trial of an accused person, as guilty or not guilty. Plutarch and other writers have stated, that in the infancy of the Greek language, they had no other word for calculating, and it then signified what is now expressed by the word áp Muziv. It is another strong presumption in favour of this supposition, that all civilised nations of the Aryan and Semitic races count by tens, tens of tens or hundreds, and so on; proceeding by multiples of ten in the successive groups of numbers.

There appears no reason why the number ten should have beer chosen rather than twelve or any other number for the scale o.

1 Odyss. iv. 412.
This most ancient manner of giving sentence is noted by Ovid :-

Mos erat antiquis, niveis atrisque lapillis,
His damnare reos, illis absolvere culpa. -Oc. Mcl. xv.

numeration, except the primitive practice of counting by the fingers of both hands. It appears highly probable that at first men counted by their fingers whatever objects did not exceed the number ten. They could also divide any multitude of objects into sets of ten in each set, and reserve what remained over when the whole multitude did not make an exact number of tens. But as the fingers could only serve to count the number of things up to ten, something else would be required to determine the number of groups of ten each ; where the number was too great for the memory to retain with ease. It appears that small stones or pebbles were used to denote them, and the vestiges of the practice among the most ancient of civilised nations has been recorded by Herodotus. It is therefore certain that in very early times, pebbles were employed in arithmetical calculations. The words calculate and calculation, being derived from the Latin calculus, a pebble, refer, in their original sense, to the use of pebbles in such operations. The Greek equivalent ψηφίζειν is obviously derived from ψήφος, 8 pebble.

It is easy to see how a multitude of objects may be divided into sets or groups; first of units less than ten, next of tens less than ten tens, and so on, by the aid of the fingers and a few pebbles.3 As soon as it was known how to number with readiness any collection of objects, it would be easy to find the number of several of these collections together, or to add them. It would be necessary first to find the sum of the groups of units, next of the groups of tens, thirdly of the groups of hundreds, and so on, and to place the pebbles in their proper places to denote the several sums of the several collections of units, tens, &c.

In proceeding to multiplication, or multiplying one number by another, it is probable that multiplication and addition at first were one and the same operation. As, for example, to multiply 12 by 4 is simply to add 12, 4 times; and in like manner to divide 48 by 12 is merely to subtract 12, 4 times from 48.

The remains of the early Greek writers on numbers are very scanty, and do not supply any adequate knowledge of the state and extent of this science in the school of Pythagoras in Magna Græcia, nor in the Academy at Athens. It appears that at a later period, B.C. 304, when Ptolemy Soter founded the great library at Alexandria, the sciences had been cultivated in that city with considerable success. Euclid promoted and advanced the sciences at Alexandria between the years B.C. 323 and 284. Some of his scien

i Herod. ii. 36.
2 Πεμπάζετ' ορθώς εκβολάς ψήρων, vou. - Æsch. Eumen. 738.

3 See Elementary Arithmetic, section vi., pp. 1-4, where has been explained and exemplifieil how any given multitude of pebbles or other objects can be divideu into groups, and named and recorded.

tific writings have descended to modern times. His chief work, “The Elements,"l combines a system of geometry and a system of arithmetic in the Greek notation, and its application to geometry. The latter is comprised in the seventh, eighth, ninth, and tenth books of the Elements.

In addition to the Elements, Pappus has recorded that Euclid was the author of a work on the Division of Surfaces and on Data, both of which are extant; also other works on Porisms, Loci, and the Conic Sections, which are lost, besides treatises on Optics and on other subjects.

To Euclid has been attributed the merit of having improved and reduced into order the principles of arithmetic and geometry which had been handed down by Thales, Pythagoras, Eudoxus, and other philosophers, and with his own additions arranged the principles of arithmetic and geometry into the form of an exact science.

Proclus affirms that Euclid more correctly ordered many parts in the Elements of Eudoxus, and completed many things in those of Theatetus, and besides confirmed such propositions as before were too slightly or insufficiently established, with the most firm and convincing demonstrations.

The mathematical school of Alexandria continued to flourish for centuries, and Alexandria was regarded as the chief seat of learning and philosophy. Almost all the eminent mathematicians and philosophers, of whose writings any remains exist, had been students at the school of Alexandria.

Of those who directed their attention to numbers, Archimedes of Syracuse, who lived between B.c. 287 and 212, improved and extended the Greek numerical notation. Apollonius adopted the scheme of Archimedes, but instead of the octads of Archimedes he adopted tetrads.

The age of Diophantus has been variously stated. Bombelli has recorded that he lived in the times of the Antonines, but upon what authority it is not evident. The more commonly received opinion is the more probable, that Diophantus lived at a later period, some time in the fourth century of the Christian era. Few facts are known of his history. An epitaph on Diophantus” is found in the third book of

1 In the year 1570, Sir H. Billingsley published a translation of Euclid's Elements in folio, with a learned preface by the famous Dr. John Dee, as he states, “Written at my poore House At Mortlake, Anno. 1570. Februarij 9." The extant works of Euclid, in the original Greek, were edited by David Gregory, Savilian Professor, and published in folio at Oxford in 1703.

2 Ούτός του Διόφαντον έχει τάφος. ά μέγα θαυμα,

Και τάφος εκ τέχνης μέτρα βίοιο λέγει.
Έκτην κουρίζειν βιότου θεός ώπασε μοίρης:

Δωδεκάτην δ' επιθείς, μήλα πόρεν χλοάειν.
Τη δ' άρ' εφ' εβδομάτη το γαμήλιον ήψατο φέγγος,


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