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No. 4, Vol. 12.] LONDON, Friday, July 29, 1825. [Price 6d.
TO WILLIAM WILLIAMS, ESQ., M. P., PROVINCIAL GRAND MASTER OF THE ASSOCIATION OF FREEMASONS FOR THE COUNTY OF DORSET.
(Concluded from page 96.) GEOMETRY, is the science of extension in all its several parts and relations of points, lines, superfices and solids. It may be divided into four classes.
First.–Rectilinear, which treats of right lines, their multipliers, combinations and proportions.
Second. Curvilinear, which treats in the same manner of circles, their divisions and intersections, as free or combined with right lines.
Third.-Trigonometry, or the properties of triangular figures.
Fourth.-Conic Sections, or the investigation of the properties and the relative proportions resulting from the formation of copi- ' cal bodies.
Hence also flow the branches of mixed and practical mathematics : among the former of which we reckon mechanics, optics and hydrostatics, or the systems of motion, light and fluids. The latter comprise almost all the arts which embellish civilized life.
Music, is the science of universal harmony; though, as an art, it is confined to the production, proportion, and combination of sounds: with respect to which, it is precisely what arithmetic is to numbers, or geometry to extension. The first great division of music is poetry, as distinguished from other productions of rhetoric by a system of measures. It consists of two branches :
First.-Prosody, or the knowledge of the measure, (i. e. the number of feet in a verse) and the time and syllables allotted to each foot.
Second.-Rythm, or the means of varying and combining the prosody, in such a manner that the several parts may form one harmonious composition. The second division of music is mechanical, that is, such as,
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producing sounds by giving modulations of voice or instruments, proportions, arranges and combines them in powerful and enchanting melody. • The ancients considered music in its more enlarged sense, as the mother of every science and the nurse of every virtue. Observing, that the laws of perfect harmony alike pervaded and combined the principles of moral and intellectual knowledge, the operations of abstract science and the laws of material essence : and hence those unerring principles, by which the system of the universe is governed, were by ihem denominated the music of the spheres-a designation which naturally leads our minds from every subordinate subject of scientific enquiry to the last and most distinguished number -the science of Astronomy, by which we are initiated into the great mysteries of the created universe, the laws which the heavenly bodies observe in their relative, motions, and particularly those of the planetary system, of which we from a part. In the first great branch of this glorious study, we consider the form, divisions, revolutions and other phenomena of the earth which we inhabit and its attendant moon. Hence we learn, to reason partly from analogy, partly from observation, on the distances, revolutions and characteristic differences of its sister planets. The fixed stars, in their slowly changing courses, their probable forms and uses, their divisions into constellations, illustrative of ancient or mythological story demand our next attention, till the excursive mind expatiating through the wonders of the unbounded universe, feeling and acknowledging the weakness of its greatest energy and the imperfection of its bigh attainments, seeks repose in the contemplation of its father and its God.
FIFTH SECTION. On ascending this staircase, the Fellow Craft was conducted to the door of the middle chamber, wbich was situated over the body of the holy house itself. When he obtained admittance by the help of a pass word and grip. The history of this pass word is found in the twelfth chapter of the book of Judges. It signifies an ear of corn springing beside a stream of water, and therafore denotes fertility, and is an impressive emblem of the first and most beneficial employment of the human faculties-the science of agriculture.
On entering the middle chamber, the Fellow Craft beheld it inscribed on every side with geometrical emblems and numerical combinations, and is instructed in the mysterious relations which they bear to the laws of the creation. In the centre, within a glorious irradiation or blazing star, is inscribed the letter G, denoting the great and glorious science of symbolical and mystical geometry, as cultivated by our ancient and venerable masters in
every age and country. The next emblem is the Triangle, generally denominated Pythagorean; because it served as a main illustration of that philosopher's system. This emblem powerfully elucidates the mystic relation between numerical and geometrical symbols. It is composed of ten points, so arranged, as to form one greater equilateral triangle, and at the same time to divide it into nine similar triangles of smaller dimensions. The first of these, representing unity, is called a MOND, and answers to what is denominated a point in geometry, each being the principle by the multiplication of which all combinations of form or number are respectively generated. The next two points are denominated a DUAD, representing the number 2, and answers to the geometrical line, which, consisting of length without breadth, is bounded by two extreme points. The three following points are called the TRIAD, representing the No. 3, and may be considered as having an indissoluble relation to all superfices, which consist of length and breadth, when contemplated as abstracted from thickness,
This relation is proved by the consideration, that no rectilinear surface can have less than three points of extension. The four points at the base, denoting the No. 4, bear a similar relation to a solid, wherein are combined the three principles of length, breadth and thickness inasmuch, as, no solid can have less than four extreme points of boundary. And, for as much as, all other abstract ideas of the point, line and superfices, are analytically derived from, and synthetically included in, that of a solid body. • The Pythagoreans affirmed the Tetractys, or number four, to be the sum and completion of all things, and the rather, also, because, in its progressive generation is completed the duad number ten- the recurring series by which arithmetical calculation is effected. The Pythagorean philosophers, therefore, considered the No. 4, first as containing a duad, which is the sum of all numbers ; secondly, as completing an entire or perfect triangle ; thirdly, as comprising the four great principles, both of arithmetic and geometry; fourthly, as representing, in its several parts, the four elements of fire, air, water and earth, and collectively, the whole system of the universe ; lastly, as separately typifying the four eternal principles of existence, generation, emanation and creation; and hence collectively, denoting the great architect of the universe. Wherefore to swear by the Tetractys was the most sacred and inviolable oath.
CLAUSE TWO. Having thus minutely examined the form and import of the Tetractys, we come next to consider some of the principal geometrical diagrams, by which we are surrounded. Let us begin with the properties of the most simple geometrical principle, the point, and proceed gradually to the relations of lines, the gene,
ration of superfices and the construction of regular solids ; but confining our enquiry to those symbols, which alone have any aptitude to mystical geometry, as being either perfect or proportinal in their several relations.
Of all geometrical points, the centre, from which a circle is generated, is the most perfect, as bearing an equal relation to every part of the circumference. Of straight lines, the most perfect relation is that of the parallel extension ; because it is by its nature immutable and interminable. Of the curved lines, the circle is the most perfect, as being in itself complete, without visible beginning or end, bearing an equal relation throughout all its parts to the generating point and containing the largest possible snperfices, within the most simple boundary of any given extent.
From the combination of the circle and right line is derived the right angled triangle, the most simple of all rectilinear superfices; for if a straight line be drawn through the centre of any circle, so extended as to touch the circumference at both extremities, and the extreme points thereof be both joined to another point of the circle, the angle found by their division will be invariably a right angled triangle, and will either be Isosceles, i. e. having the sides which include the right angle equal-or Scaleni, i. e. having all its sides and angles unequal. The former of these possesses the capacity of infinite reduplication and may also be infinitely divided into similar triangles, equal to each other, observing in both respects, the geometrical progression founded on the duad or No. 2, and in every such operation, the whole as well as the parts still retaining its original characters, form and relation.
In its Scaleni conformation it is in like manner divisible, and its divisions retain their former proportions and relations ; but if multiplied, it becomes the basis of the trilateral forms, which vary according to the proportions of its angles and the combination of its lines.
When two Scaleni right angled triangles of equal dimensions, are united by the smallest of the lines which include the right angle, they form an obtuse angled triangle of the Isosceles order: when, by the larger of these two lines, an acute angled triangle of the same description. But in the latter case, their angles are to each other, in the arithmetical proportion of one, two and three. They form an equilateral triangle, which may be justly considered as the most perfect of all trilateral forms, for the following reasons :-first, because, it is equal in all its relations : second, because, it is capable of being reduced into right angled Scaleni and obtuse isosceles: thirdly, because it is infinitely divisible, or may be infinitely multiplied, into similar triangles, equal to each other, without alteration of its form or relations : fourthly, because in every such division or augmentation, it observes the geometrical progression founded on the tetrad or No. 4; and, therefore, it may
be considered a symbolical representation of that species of proportion.
Of quadrilateral superfices, the most simple is the square, formed by uniting the hypothenuse or side subtending to the right an. gle of two right angled Isosceles triangles, containing equal. It is also most perfect on account of the equality of its relations in the same manner.
The rectangular parallelogram is founded by the similar union of two scaleni triangles of the same description.
A rhomb is the union of two equilateral triangles. A rhomboid of two right angled triangles, conjoined by the larger of those sides which contain the right angle; but in an inverted position.
Of trilateral and quadrilateral figures, it is to be observed, that none are admissible into symbolical geometry, but those which, in their respective lines and angles, bear the relation of equality or such integral proportions, as may be adequately expressed by some of the numerical terms of the letractys, i. e. the numbers 1, 2, 3, 4,
We next proceed to the construction of multilateral figures. Having their sides and angles equal, these are invariably formed by the combination of as many acute angled triangles, as the figure has sides.- This class of forms may be sufficiently illus. trated by the pentagon, which resolves itself into five isosceles acute angled triangles; but there is one which requires particular notice; I mean the hexagon, which, being composed of six equilateral triangles, is equal in all its relations, and retains the quality of being infinitely divisible into similar triangles, according to the geometrical projection observed in the divisions of that trilateral figure, and may, therefore, be considered as the most perfect of all multilateral forms.
From this enquiry, it results, that the three most perfect of all geometrical diagrams are the equilateral triangle, the square, and the equal hexagon. To this we may add an observation, for which we are indebted to our grand master Pythagoras, that there exists no other regular equilateral forms, whose multiples are competent to fill up and occupy the whole space about a giveu centre: which can only be effected by six equilateral triangles, four squares, and three equal hexagons. There are but five regular solids contained under a certain number of equal and similar superfices, which, from the use made of them in the Platonic philosophy, are usually denominated the five Platonic bodies. Those are,
A Tetracdon, or pyramid, contained under four equal and equilateral triangles, representing, according to the Platonists, the element of fire.
An Octaedron, contained under eight such triangles, represents air.