The Pythagorean Triangle.*

Pythagoras on the combinations of the point , the line , the superfice , and the solid , to form rectangular , trilateral , quadrilateral , multilateral figures , and the regular bodies ; the latter of which , on account of their singularity , and the mysterious nature usually ascribed to them , were formerly known by the name

of the five Platonic bodies ; and they were so highly regarded by the ancient geometricians , that Euclid is said to have composed his celebrated work on the elements , chiefly for the purpose of displaying some of their most remarkable properties . These disquisitions usually conclude with an explanation of the

fortyseventh Problem of Euclid , which is commonly called the Eureka of Pythagoras . It appears to me that in the revision of the English ritual at the union , a great omission was perpetrated , which it would be well to supply ; and in the present taste for scientific lectures and investigationsnothing

, would tend to elevate the character of Freemasonry more than to afford an opportunity for its indulgence by furnishing the means of carrying out its references by the introduction of a higher range of science . Freemasonry , to be completely successful , should take precedence in science as it does in morals and the

exercise of charity ; for there are few institutions which equal it hi the walks of benevolence . It is true the seven liberal sciences are referred to in the second degree ; but with the exception of geometry , they occupy no important place in the ritual . And for this reason I suppose , that in ancient times the Order was denominated Geometry . On this science , with its application to architecture , our disquisitions are abundant and powerfully interesting ;

and why should not a lecture on the elementary principles of other sciences be equally gratifying to the members of a lodge ? Arithmetic , or the science of number , is nearly allied to geometry ; we patronise music in practice , but hear little of it in theory ; and of astronomy we are merely told that it is an art which teaches us to read the wonderful works of God

in those sacred pages the celestial hemisphere , where we may perceive unparalleled instances of wisdom and goodness , and through the whole of the creation trace the glorious Author by his works . That great philosopher Pythagoras , who , by the superiority of his mindinfused a new spirit into the

, science and learning of Greece , where he founded the Italic sect , taught his disciples geometry , that they might be able to deduce a reason for all their thoughts and actions , and to ascertain correctly the truth or falsehood of any proposition by the unerring process of mathematical demonstration . Thus being

enabled to contemplate the reality of things , and to detect imposture and deceit , they were pronounced to be in the road to perfect happiness . Such was the discipline and teaching of the Pythagorean lodges . And it is related that when Justin Martyr applied to a learned Pythagorean to be admitted as a candidate

for the mysterious dogmata of his philosophy , he was asked whether , as a preliminary step , he had already studied the sciences of arithmetic , jnusie , astronomy , and geometry , which were esteemed to be the four divisions of the mathematics ; and he was told that it was impossible to understand the perfection of beatitude without them , because they alone are able to abstract the soul from sensibles , and to prepare it for intelligibles . He was further told that in the absence

of these sciences no man is able to contemplate what is honest , or to determine what is good ; and because the candidate acknowledged his ignorance of them , he was refused admission into the society . Above all other sciences or parts of the mathematics , the followers of Pythagoras esteemed the doctrine of Numberswhich they believed to have been revealed

, to man by the celestial deities ; and they pronounced arithmetic to be the most ancient of all the sciences , because , "being naturally first generated , it takes away the rest with itself , but is not taken away with them . Thus animal is first in nature before man , and btaking away animal we take away manbut b

y , y taking man we do not take away animal . " They considered number extending to the deead to be the cause of the essence of all things , and therefore esteemed the creation of the world to be nothing more than the harmonious effect of a pure arrangement of number . Again , the monad was esteemed by Pythagoras as

the father of number , and the duad as its mother ; whence the universal prejudice in favour of odd numbers , the father being had in greater honour than the mother . Odd numbers being masculine were considered perfect , and applicable to the celestial gods ; while even numbers , being female , were considered

imperfect , and given to the terrestrial and infernal deities . Every tyro knows that odd numbers are Masonic ; and if he be ignorant of the reasons why three , five , seven , and eleven have been adopted as landmarks , let him apply to the Master of his lodge for information , and he will then be satisfied of the wisdom of the appropriation , because number forms one of the pillars which contribute to the support of scientific Masonry , and constitutes an elementary principle of geometry .

Thus , in the Pythagorean triangle consisting of ten points , the upper dot or jod is monad , or unity , and represents a point , for Pythagoras considered a pointto correspond with unity '; a line to two , a superfice to three , a solid to four ; and he defined the point as a

monad having position and the beginning of all things ; a line was thought to correspond with duality , because it was produced by the first motion from indivisible nature , and formed the junction of two points . A superfice was identified with the number three ,, because it is the first of all causes that are found in fi for a circlewhich is the principal

gures ; , of all round forms , comprises a triad , in centrespace—circumference . But a triangle , which is the first of all rectilineal figures , is included in a ternary , and receives its form according to that number ; and was considered by the Pythagorians to be the author of all sublunary things . The four points at the base

of the Pythagorian triangle represent a solid or cube , which combines the principles of length—breadththickness ; for no solid can have less than four extreme boundary points . Thus it appears that in applying number to physical things , the system of Pythagoras terminated the

in a tetrad ; whilst that of Aristotle , by omitting point , limited the doctrine of magnitude to the triad of line—surface—body . In divine things , however , the former philosopher profusely used the number three , because it represented the three principal attributes of the deity . The first of which is infinite with fecundity ; the second , infinite " knowledge and