User:Robert.Baruch/Opuscula Mathematica

1625.

By the Reverend Doctor Francesco Maurolico, Abbot of Messina and esteemed mathematician.

On the Sphere, Book one.

To the reader: A prologue.

On the Sphere, and On the Computation of Times consists of the writings of many men having written long ago: chief among them are Johannes de Sacrobosco, Robert bishop of Langres, and Campanus. It is pleasing, however, to here touch upon a few things whatever pertains to this kind of business. Indeed the treatise of the Spheres contains the rudiments of astronomy: computation of the calendar, the feasts of the month and the calculation of the year, as if a certain introduction to calculation. For this reason, as the Sphere is seen to be a certain theory of motion, so the Computation is the practice and reckoning of the same. Each of them comes in use for the convenience of the general public. Derived from each is the discrimination of feasts, times, lunations and celebrations, as well as the series of years according to Consuls, Popes, Kings, and the histories of successions and the reckoning of daily affairs and uses in business matters. Thus we shall touch upon these principles and precepts to add. I shall perhaps supplement some omission from others, or, if the work is superfluous, I shall curtail it. In both, the authors prattle on. The teaching is more easily grasped when shorter. He who does not know Campanus spoken so fully, as in the Sphere, so in The Computation, not for the necessity of business, but for showing off: and so that he may equally argue Sacrobosco in ignorance? But if only he himself had not gone astray, excessively confident in his acumen in the teaching of the elements of Euclid, he would not have sensed his revision of Johannes Regiomontanus. But these things will be struck down elsewhere.

On the division of knowledge

Since all knowledge dwells around the subject about which it discusses: it is in the subject, that is in the mind: it is said about the subject, like the general about the particular. For that reason philosophy can be distinguished in three ways. First, according to the division of subject, such as into Organics and reals, like a subject into a sign and the thing signified. Second, according to the objects of the mind's abilities: as into the consideration of truth, which is the object of intellect, and morals, which is the object of will. Third, according to the division of kinds in types: as into Theory and Practice, which are the two main types of Philosophy.

In addition, knowledge precedes knowledge in five ways. First, as the general precedes the particular. For example, Philosophy to universals, Mathematics to Geometry, Logic to Grammar. Second, according to the order of discovery, like the thought of the particular precedes the thought of universals: when we proceed from the sense to the intellect. Third, in the process of learning, or teaching: like Grammar precedes Rhetoric and Dialectics, and so on. Fourth, by the nobility of the subject: like Theology precedes Astronomy, and Astronomy precedes Geography. Fifth, by the certitude of demonstration, like Geometry precedes Astronomy, and Astronomy precedes Physics.

According to the division by subject:

Philosophy	Organics or Logics	Grammar	Poetry
			History
		Rhetoric
		Dialectics
	Reals	Speculative	Metaphysics
			Mathematics	Arithmetic
				Geometry
			Physics	Music
				Astronomy
		Practical	Active
			Factive

According to the objects of abilities:

Philosophy around the object of intellect or will	Truth around speculation	In one ?	Metaphysics
			Theology
		In two ?	Nature	Mathematics
				Physics
			Art	Logic	Grammar
					Rhetoric
					Dialectics
				Arithmetic	Music
					Measurement
				Geometry	Astronomy
					Geography
	Moral good around practice	Of ? around customs to be cultivated
		Of the body around mechanical exertion

Thus through the triple respect, Philosophy can be distinguished in three ways. May the diversity of positions not acutely disturb you, reader: since in each of the three divisions of knowledge and arts (however they may be arranged) stand always mutually connected, and propagated from the same root.

Now, the Speculative part of Philosphy is divided into the natural, mathematics, and theology. Theology proceeds from the human faculty. Physics, because of the flux of material, is not stable. For that reason mathematics, and chiefly astronomy, is recommended because of their certitude of demonstration and nobility of subject, as Ptolemy said. Therefore the principles of astronomy about to be taught, we first teach certain necessary and geometric prefaces.

The geometric principles

A point is a mark without quantity. A line is length lacking width. Also of lines, some are straight, others curved. A surface is that which has length and width. Of surfaces, some are planar, others curved. An angle is the indirect coming together of lines. Of angles, some are rectilinear, others in other ways. A straight line is perpendicular to another straight line, when either makes equal angles, they are called right angles. An angle greater than right is called obtuse; less is called acute. A terminal is a limit or a border. A shape is that which is closed by a terminal or terminals. A circle is a shape whose center is equally distant from its periphery: a diameter drawn through the center divides it into semicircles: a straight line beside the center divides it into unequal portions. Of the rectilinear shapes certain ones are trilateral, certain ones are quadrilateral, others are multilateral. Now of the triangles, certain are equilateral: certain are Isosceles, which has only two sides equal: certain are Scalene, which has three unequal sides. Likewise, some are orthogonal, where one of the angles is a right angle: others are amblygonal, where one is obtuse: others are oxygonal, which has all acute. Always, however, the largest side is opposite the largest angle. And the three angles joined together make two right angles. There are five kinds of quadrilateral shapes: the square, rhombus, rectangle, rhomboid, and trapezium. Only the first and second of these have equal sides, and the first and third have right angles. The second and fourth have opposite angles equal. The third and fourth have opposite sides equal. The last is neither equilateral nor equiangular. Now parallel, or equidistant, straight lines are those having been drawn in the same plane and do not contact no matter how long or in what way they are extended. A parallelogram is that of which opposite sides are equidistant. The first four types of equilaterals are of such a sort. The limits and intersections of lines are points. Of whatever rectilinear shapes are chosen, all their angles joined make up in total as many pairs of right angles as triangles they are divided into. From where the four angles of a quadrilateral shape forge four right angles, because it is resolved into two triangles. The angles of a pentagonal shape forge six right angles, because it is cut into three triangles, and so on and so forth.

A solid is a body contained under the triple dimension, that is, what is length, breadth, and depth. A perpendicular line into a plane is that which makes right angles with straight lines having been drawn on the plane. Parallel planes are those which never meet no matter how long and no matter in what direction they are extended. Parallelepiped solids are those whose opposite bases are parallel. Types of solids are pyramids, columns, prisms, and polyhedral shapes. A solid angle is made from the meeting of three or more plane angles, it is necessary that four of these are less than right angles. Of the polyhedral shapes there are five such solids which are called regular, since they are each bounded under equilateral, equiangular, and equal bases within themselves. The pyramid has four triangles. The octahedron, eight. The cube has six squares. The icosahedron has twenty triangles. The dodecahedron has twelve pentagons. A cone is a round pyramid above a circular base. A cylinder is a round column having as bases equal and parallel circles. In these, an axis is led through the peak and centers of the bases. When the axis is perpendicular to the base, it is called a right cone and right cylinder. Now, cut, it is scatenus. As two straight lines cutting each other mutually, so too does every rectilinear triangle lie in one plane. A sphere is a solid enclosed by one surface from which the center in the middle is equally distant. Its diameter or axis goes through the center, as Theodosius says. Or as Euclid says, a sphere is a solid which is described by a semicircle revolved around a fixed diameter. A ratio, or proportion, is a comparison of quantities of the same kind. Similar, same, or equal ratios are those which are either of the same name, or is simultaneously bigger by whatever named ratio, or simultaneously smaller. Now the ratio is named by numbers. Quantities of the same ratios are called proportional.

Similar planar and similarly positioned shapes are those whose each and every angle is equal and all the same. And, each and every side is proportional and equidistant. Similar and similarly located solids are those contained under similar, the same number of, and parallel bases. And it is sometimes possible to make it so that two sides of shapes in similar position of planes, or pairs, simultaneously are congruent. And in similar location of solids, two bases or a pair or a triple may unite on one plane, with the rest equidistant. Correlative sides or correlative bases, each and every correlative to be collected. Likewise, similar cones or similar cylinders are those whose axes are proportional with respect to the diameter of the bases, and are inclined rightly or equally. Now every two circles and every two spheres are mutually similar, since they always have proportional diameters of their perimeters. Likewise, in circles, proportional chords of the diameter cut off similar portions, which receive equal angles positioned either to the center or to the periphery. Also in spheres, similar circles (the diameters of which are proportional to the diameter of the sphere) cut off spherical portions. Now, just as a parallelogram is double its triangle, a tetragonal column is double its Serratile. Likewise, just as a column is triple its pyramid, a cylinder is triple its cone. Likewise, two triangles, two parallelograms, two columns, two pyramids, or cones constituted on equal bases are proportional in their peak. But if they are of the same height, they are proportional in their bases. Likewise, the angles in circles, whether it terminates at the center or at the periphery, are proportional in their received peripheries.

Now, similar planar shapes are in double ratio of their responding sides. Thus also two circles are in double ratio of their diameters. And similar solids are in triple ratio of their diameters. Now in other shapes, whether triangles on planes or parallelograms you may bring together, or in solid pyramids or parallelepipeds or columns you may bring together, the ratio of the shapes brought together is always constructed from the ratios of the bases and heights. From there, if the bases are reciprocal with respect to the heights, the shapes must be equal, and vice-versa.

Now, when a straight line cuts two parallel lines, the oppositely positioned angles, which are reciprocal of each other, and externally and internally opposite, are mutually equal. The two internal angles taken together are equal to two right angles. And one position of these makes it equidistant. When a straight line cuts a straight line, the two collateral angles are either right angles or equal to two right angles. And all four angles are either right angles or all together equal to four right angles. From there, four squares together, or three equiangular hexagons, or six equilateral triangles can fill up all space with angles meeting. Since, of course, in a square, the measure of the angle is a right angle. In a hexagon, it measures one and one third right angles. In a triangle, it measures two thirds of a right angle. And for that reason just as the four angles of a square, and the three angles of a hexagon, and the three angles of a triangle, four right angles must be equivalent. Likewise, if you consider the quantity of angles, like in hexagons with intermixed triangles, or octagons with intermixed squares joined together, you will enclose the total area by such a ratio as experiment. Now, these things are to be pre-tasted from the elements of Euclid and pre-studied by those who wish to pursue the astronomical principles. But also the spherical elements of Theodosius are by no means to be omitted, so that the form of the universe-sphere, the magnitude of its circles, its positions, inclinations, axes, poles, and division may be understood.

If a plane cuts a sphere through the center, the section will be a great circle having its center in common with the sphere, and the plane will be cutting the same sphere into two hemispheres. Now if a plane cuts a sphere next to the center, the section will be a small circle, having a center outside the center of the sphere, and cutting the sphere into two unequal portions. From there, all great circles in a sphere are mutually equal, and mutually divide themselves into semicircles since they have a common center. Now, small circles equally remote from the center of the sphere are equal. A more remote circle is less. The axis of a sphere is its diameter on which it moves. And it is its circles through whose center it transits perpendicularly. The poles are the extremes of the axis which are each equally removed from the periphery of its circle. Parallel circles in the sphere have the same axis and poles and vice-versa. Take a great circle in the sphere drawn through the poles of equidistant circles: it divides them each through equality. Now, if it is next to the poles, through unequal (excepting the equidistant major ones) arcs coalternating of two circles both equally remote from the middle, are equal. The farther away, the greater the inequality. Likewise, it makes more oblique cuttings. A great circle drawn through the poles of circles mutually cutting each other in the sphere divides both portions of them equally, and of those mutually touching each other in the sphere, goes through the contact point. If two great circles go through the poles of equidistant circles, whether they touch a minimum of them, then their arcs enclosed between the great semicircles are similar, and the arcs of the great circles included by the two equidistant circles are equal.

With these premises we shall come to the introduction of the universe-sphere. Now, whatever is to be taught over and above that matter, whether it pertains to the principles, or to circles, or the prime motion, or secondary motions, each of these we shall set forth cursorily and in brief.

The principles of the sphere, which are the six conclusions of Ptolemy

That the shape of the heavens is spherical, and its motion is circular. For with the heavens about to grasp everything suitable, it made the most capacious shape for circular motion that can be made, and which might always contain boundaries between themselves, and such is a spherical shape. Likewise if it were cut, the heavens because of its circular motion would be broken, or vacuum would be found. This very thing is perceptively proven by experiment. Now, it stands that the bodies of the stars are spherical since, in whichever direction they are considered, they seem round. Likewise by necessity, their motion by example of the heavens and the elementary forms, by the rising and falling of the Moon.

That the Earth is round. For the anticipation, from rising to setting, of the rising and setting of the stars through evidence of the lunar eclipse proves its roundness. The increase of the heights of the meridians and poles of the universe from south to north indicates its roundness. Now, that such roundness is circular, is attainable, because of the said anticipations, the increases are proportional with respect to the spaces of the locations. That the water is round, is shown by the said argument. Likewise, by the successive appearance of crags, hilltops, and islands. That the entire globe is round, is shown similarly. Likewise from the shadow of the Earth in the absence of sunlight of the Moon. From the equal pressure in the center, and the equal distance from the center. That the entire universe is round, is proven by the similarity of the universe archetype. However, the land because of its hardness cannot acquire perfect roundness, but the prominences of mountains or valleys are not perceived compared to such softness.

That the Earth is situated in the center of the universe. Indeed, this follows, since we see semicircles from the whole heaven of hemispheres, from great circles, and the increments of the days and nights do not correspond otherwise, nor the lunar eclipses, nor are the lines of equinoctial shadows defined in straight lines. Likewise, since it is shown that it is in two diameters of the universe, it follows that it is in the center. The same begs a serious law of nature of compelling it into the middle.

On the universe

The universe is a sphere, whose center is that of everything and of the Earth. Its rounded surface is that of the prime mover, or that of the farthest heavens. Now, since a sphere is a solid, and a solid is closed by a surface or surfaces, a surface is bounded by a line or lines, and a line lies between points, for that reason snatching these foundations of astronomy are to be mentioned before, indeed not only geometry but also the precepts of arithmetic are to be examined.

On axis and pole

Now the diameter of the axis of the universe is of this spherical machine, above which the sphere itself, or Universe, turns from rising to setting. The poles are the extreme points of the axis. From there all astral bodies, all stars, indeed everything, whatsoever, is in the world, their points are carried in a circular motion, and describe great circles