Plutarch's morals. Part 2. translated from the Greek by several hands.

A Breach in the Original.

But the Quaternary of Numbers set down by 〈 math 〉〈 math 〉 Plato have a more perfect Generation, of even Numbers multiplyed by even distances; of odd, by une∣ven Intervals. This Quater∣nary contains the Unite, the common Original of all even and odd Numbers. Subsequent to which are two and three, the first plain Numbers, then Four and Nine, the first Tetragonals; and next Eight and Twenty seven, the first Cubical Num∣bers, substracting the Unite from the rest. Whence it is apparent, that his Intention was not that the Numbers should be plac'd in a direct Line, one above another, but a part, and oppositely one against t'other, the even by themselves, and the odd by themselves, accord∣ing to the Schemes in view. In the same manner are similar Numbers likewise to be joyn'd toge∣ther, which will produce other Numbers re∣markable, as well by their Addition, as Multi∣plication of one another. By Addition thus, two and three make five, four and nine make thirteen, eight and twenty seven, thirty five. Of all which Numbers the Pythagoreans call'd Five the Nourisher, that is to say, the Breeding or

Fostering sound: believing a Fifth to be the first of sounds, expressing the Intervals of a Tone. But as for Thirteen, they call it the Remainder, dispair∣ing, as Plato himself did, of being ever able to divide a Tone into equal parts. Then Five and Thirty they nam'd Harmony, as consisting of the two first Cubes, rising from an odd and an even Number; as also out of the four Numbers Six, Eight, Nine and Twelve comprehending both Harmonical and Arithmetical Proportion. Which nevertheless will be more conspicuous being made out in a Scheme to the Eye.

Admit a Right Angle Parallellogram, 〈 math 〉〈 math 〉 A.B.C.D. the lesser side of which A. B. consists of Five, the longer side A. C. contains seven Squares. Let the lesser Division be unequally divided into two and three Squares, mark'd E. And the larger Division into two unequal Divisi∣ons more of three and four Squares, mark'd F. Thus A.E.F.G. com∣prehends six, E B.G.I. nine. F.G.C.H. eight, and GIHD. twelve. By this means the whole Paral∣lellogram containing thirty five little square A∣reas, comprehends all the Proportions of the first concords in Musick in the number of these little Squares. For six is exceeded by eight in a Sesqui∣terce proportion, wherein the Diatessaron is compre∣hended. And six is exceeded by nine, in a Ses∣quialter proportion, wherein is also included the fifth. Six is exceeded by Twelve in duple Pro∣portion, containing also the Octave; and then

lastly, there is the Sesquioctave Proportion of a Tone in eight to nine. And therefore they call that Number which comprehends all these Propor∣tions, Harmony. This Number is 35, which being multiply'd by 6, the Product is 120. Which is the Number of Days, they say, which brings those Infants to Perfection that are born at the seven Months end. To proceed by way of Multiplication, twice 3 makes 6. And four times 9 thirty six, and 8 times 27 produces 216. Thus 6 appears to be a perfect Number, as be∣ing equal to its Parts, and therefore called Ma∣trimony, by reason of the Mixture of the first Even and Odd. Moreover, it is compos'd of the O∣riginal of Number, which is One, of the first even Number, which is Two, and the first odd Num∣ber, which is Three. Then for 36, it is the first, as well Quadrangular as Triangular Number. Quadrangular from 6, and Triangular form 8. The same thing happens from the Multiplication of the two first square Numbers, 4 and 9, as al∣so from the Addition of the three Cubical Num∣bers. One, Eight, and 27, which being put to∣gether make up 36. Lastly, you have the une∣qual sides of the Parellellogram, by the Multipli∣cation of 12 by 3, or 9 by 4.

Take then the Numbers of the sides of all these Figures, the 6 of the Square, the 8 of the Triangle, the 9 for the one side of the Parallello∣gram, and the 12 for the other side, and there you will find the Proportions of all the Concords. For 12 to 9 will be a Fourth, as De la sol re to A la mire below. To 8 it will prove a Fifth, as De la sol re to G sol re ut below. To six it will be an Octave, as D la sol re to D solre. And the two hundred and sixteen is the Cubical Number, pro∣ceeding

from six, which is its Root, and so e∣qual from the Senarie to its own Perimeter. Now these Numbers aforesaid being endu'd with all these Properties, the last of them, which is 27, has this peculiar to it self, that being added, to those that preceded, it is equal to All together; besides that, it is the Periodical Number of the Days wherein the Moon finishes her monthly Course, the Pythagoreans make it to be the Limit of all Harmonical Intervals. On the other side, they call Thirteen the Remainder, in regard it misses a Unite to be half of Seven. Now that these Numbers comprehends the Proportions of Harmoniacal Concord, is easily made apparent. For the Proportion of 2 to 1 is duple, which contains the Diapason; as the Proportion of 2 to 3 Sesquialter; which embraces the Fifth: and the Proportion of 4 to 3 Sesquiterce, which com∣prehends the Diatessaron. The Proportion of nine to three Triple, including the Diatessaron and Diapente, and that of 8 to 2 Quadruple, com∣prehending the double Diapason. Lastly, there is the Sesquioctave in 8 to 9, which makes the Tone Major, counting then the Unite which is com∣mon as well to the even as the odd Numbers, the whole Series of Figures compleats the Decad. For the first four Numbers from the Unite, 1, 2, 3, 4, make Ten: and these even Numbers, 1, 2, 4, 8, produce 15, in order the third Triangular or Trigonal Number from Five. On the other side, take the odd Numbers, 1, 3, 9, and add to them 27, the product is 40, by which Numbers the Skilful measure all musical Intervals, of which they call'd the one a Diesis (or the half of a Semitone Minor) and the other a Tone. Which Number of 40 proceeds

from the force of the Quaternary Number by Multiplication. For from the first four Num∣bers, every one being multiply'd four times by it self, the Product will be 4, 8, 12, 16, which being added altogether make 40, comprehending all the Proportions of Harmony. For 16 is a Sesquiterce to 12, Duple to 8, and Quadruple to 4. Again, 12 holds a Sesquialiter proportion to 8, and Triple to 4. In which proportions are contain'd the Intervals of the Diatessaron, Diapente, Diapason, and double Diapason. Moreover, the Number 40 is equal to the two first Tetragones, and the two first Cubes being taken both together. For the first Tetragones are 1 and 4, the first Cubes are 12 and 27, which being added toge∣ther make 40. Whence it appears that the Pla∣tonic Quaternary, is much more perfect and fuller of Variety than the Pythagoric; but in regard the Numbers propos'd did not afford space sufficient for the middle Intervals, therefore there was a Necessity to allow larger Bounds for the Propor∣tions.

And now we are to tell ye what those Bounds and middle Spaces are. And first concerning the Medieties; of which, that which equally exceeds and is exceeded by the same Number, is call'd Arithmetical; the other which exceeds, or is ex∣ceeded by the same part of its Extremities, is call'd Subcontrary. Now the Extreams, and the middle of Arithmetical Mediety are 6, 9, 12. For 6 is exceeded by 9, as nine is exceeded by 12, that is to say, by the Number three. The Ex∣treams of the Subcontrary are 6. The Extreams and middle of the Subcontrary are 6, 8, 12, where 6 is exceeded two by 8, and 8 four by 12, yet 2 is equally the Third of 6, as 4 is the third

part of 12. So that in the Arithmetical Mediety, the Middle exceeds and is exceeded by the same part; but in the Subcontrary Mediety, one of the Extreams wants, the other abounds in the same part of the Extremity, for in the first, 3 is the third part of the Medium in reference to both Ex∣treams; but in the latter, the third parts are dif∣ferent, 4 and 2, whence it is call'd Subcontrary. This they also call Harmony, as being that whose Middle and Extreams afford the first Concords: that is to say, between the highest and lowermost lies the Diapason: between the highest and the middle lies the Diapente; and between the mid∣dle and lowermost lies the Fourth or Diatessaron. For suppose the higest Extream to be D la sol re, and the lowest Extream De sol re, the middle is G sol re ut, making a Fifth to the uppermost Ex∣tream, but a Fourth to the lowermost. So that D la sol re answers to 12, G sol re ut to 8, and D sol re to 6. Now the more readily to find out these Mediums, Eudorus hath taught us an easie Method. For after you have propos'd the Ex∣tremities, if you take the half part of each, and add them together, the Product shall be the mid∣dle alike both in Duple and Triple Proportions, in Arithmetical Mediety. But as for Subcontrary Medi∣ety, in duple proportion, first having fix'd the Extreams, take the third part of the lesser, and the half of the larger Extream, and the Addition of both together shall be the middle. In triple proportion the half of the lesser, and the third part of the larger Extream shall be the Mediety. As for Example, in triple proportion let 6 be the least Extream, and 18 the biggest; if you take 3, which is the half of 6, and 6 which is the third part of 18, the Product by Addition will

be 9, exceeding and exceeded by the same parts of the Extreams. In this manner the Mediums are found out. Now these Mediums are so to be dispos'd and plac'd to fill up the duple and triple Intervals. For of these propos'd Numbers, some have no middle Space, others have no sufficient. Being therefore so augmented that the same Pro∣portions may remain, they will afford sufficient space for the foresaid Mediums. To which pur∣pose, instead of a Unite, they choose the Num∣ber six, as being the first Number including in it self a half and third part, and so multiplying all the Figures below it and above it by 6, they made sufficient room to receive the Medium both in double and triple Distances, as in the Ex∣ample.

12	2	6	3	18
24	4	9	54
48	8	27	162

Now Plato having laid down this for a Positi∣on, that the Distances of Sesquialters, and Sesqui∣terces, and Sesquioctaves being once found out, all the Sesquiterce Distances were fill'd up from those Connexions, in the Sesquioctave Intervals, by leav∣ing such a part of each, so as the Distance left of the part might bear the proportional Extreams of Number to Number, as 256 to 243. From hence they were constrain'd to enlarge their Number and make them bigger, that there might be two Numbers following in order in Sesquioctave proportion; the six not being sufficient to con∣tain two Sesquioctaves, though you should bruise it into ten thousand Unites, which would strangely perplex the Study of these things, Therefore

the Occasion it self advis'd Multiplication. As in the Musical Scale, the Change and Variation of Notes extends it self upward and downward from the first innumerical Proportions of the Base, Eudorus therefore imitating Cranter, made choice of 384 for his first Number, being the product of 64 multiply'd by 6, which way of proceeding the Number 64, lead them to, containing it's under Sesquioctave 9, in proportion to 72. But it is more agreeable to the Words of Plato to introduce the half. For the remainder of that will bear a Sesquioctave proportion in those Numbers which Plato mentions of 256 to 248, making use of 192, for the first Number. But if the same Number be made choice of doubl'd, the Overplus or Default will have the same pro∣portion as the doubl'd Number 512 to 484. For 256 is in Sesquiterce proportion to 192, as 394 to 512. Neither was Cranters Reduction of proportions to this Number without reason, which made his Followers willing to pursue it. In regard that 64 is both a Tetragon from the first Cube, and a Cube from the first Tetragon, and being multiplied by 3, the odd and Trigonal, and the first perfect and Sesquialter Number it produ∣ces 192, which also contains its Sesquioctave, as we shall demonstrate. But first of all we shall better understand what this Leimme or Remain∣der is, and what was the Opinion of Plato, if we do but call to mind what was frequently ban∣dy'd in the Pythagorean Schools. For Interval in Musick is all that Space which is comprehended by two sounds, vary'd either by raising the Voice, or scruing the String. Of which Intervals that which is call'd a Tone is the full excess of Dia∣pente above a Diatessaron: And this being divided

into two parts according to the Opinion of the Musitians, makes two Intervals, both which they call a Semitone. But the Pythagoreans despairing to divide a Tone into equal parts, and therefore perceiving the two Divisions to be unequal, they call'd the lesser Leimme or Defect, as being lesser then the half. Therefore some there are who make the Dratessaron, which is one of the Con∣cords, to consist of two Tones and a half; o∣thers of two Tones and a Leimme. In which Case, Sense seems to govern the Musitians, and Demonstration the Mathematicians. The proof by Demonstration is thus made out. For that it is certain from the practice of Instruments, that the Diapason has double proportion; the Diapente a Sesquialter; the Diatessaron a Sesquiterce, and the Tone a Sesquioctave proportion. Now the truth of this will easily appear upon examination, by hanging two Weights double in proportion to two Strings, or by making two Pipes of equal hollowness, double in length, the one to the o∣ther. For the bigger of the Pipes will yield the deep sound, as D sol re, to D la sol re: and of the two Strings that which is extended by the double weight, will be acuter then the other, as D la sol re to De sol re: And thus much for a Dia∣pason. In the same manner two Longitudes or Ponderosities being exceeded or extended by three will produce a Diapente; and four by three will yield a Diatessaron: of which, the one carries a Sesquiterce, the other a Sesquialter Proportion. But if the same inequality of weight or length be so ordered, as Nine to Eight, it will produce a To∣niac Interval, not perfect Concord, but Harmo∣nical enough: in regard the Strings being struck one after another, will yield so many musical and

pleasing Sounds; but altogether a dull and un∣grateful Noise. But in Consort being touched either singly or together, thence a delightful Me∣lody will charm the Ear. Nor is all this less de∣monstrable by Reason. For in Musick, the Di∣apason is compos'd of the Diapente and Diatessa∣ron. But in Numbers, the Duple is compounded of the Sesquialter and Sesquiterce. For 12 is a Sesquiterce to 9, but a Sesquialter to 8, and a Du∣ple to 6. Therefore is the duple proportion compos'd of the Sesquialter and Sesquiterce, as the Diapason of the Dia••ente and Diatessaron. For here the Diapente exceeds the Diatessaron by Tone, there the Sesquialter exceeds Sesquiterce by a Sesquioctave. Whence it is apparent that the Diapason carries a double Proportion, the Diapente a Sesquialter, the Diatessaron a Sesquiterce, and the Tone a Sesqui∣octave. This being thus demonstrated, let us see whether the Sesquioctave will admit a Division in∣to two equal parts; which if it will not do, nei∣ther will a Tone; however, in regard that 9 and 8, which make the first Sesquioctave, have no middle Interval; but being both multiply'd, the Space that falls between causes two Intervals, thence it is apparent, that if those Distances were equal, the Sesquioctave also may be divided into equal parts. Now the double of 9 is 18, of 8, 16; the Intermedium 17; by which means one of the Intervals becomes larger, the other lesser; for the first is from 18 to 17, the second is from 17 to 16. Thus the Sesquioctave proportion not being to be otherwise then unequally divided, consequently neither will the Tone admit of an equal Division. So that neither of these two Sections of a divided Tone are to be call'd a Se∣mitone, but according as the Mathematicians

name it, the Remainder. And this is that which Plato means, when he says, that God having fill'd up the Sesquiterces with Sesquioctaves, left a part of each: of which the Proportion is the same, as of 256 to 243, the remainder being 1 10/243. For admit a Diatessaron in two Numbers, compre∣hending a Sesquiterce proportion, that is to say, in 256 and 192: Of which two Numbers, let the lesser 192 be apply'd to the lowermost Ex∣tream, and the bigger Number 256, to the up∣permost Extream of the Tetrachord. Whence we shall demonstrate, that this space being fill'd up by two Sesquioctaves, such an Interval remains as lyes between the Numbers 256 and 243. For the String being forc'd a full Tone downward, which is a Sesquioctave, it makes 216, but being screw'd a full Tone upward, it makes 243. Which 243 exceeds 216 by 27, and 216 exceeds 192 by 24. And then again, of these two Numbers, 27 is a Sesquioctave to 216, and 24 the Sesqui∣octave to 192. So the biggest of these two Numbers is a Sesquioctave to the middle, and the Middle to the least; and the distance from the least to the bigest, that is, from 192 to 243, consists of two Tones fill'd up with two Sesqui∣octaves. Which being substracted, the remain∣ing Intervals of the whole between 243 and 216 is 13, for which reason they call'd this Number the Remainder. And thus I am apt to believe the Meaning and Opinion of Plato to be most exactly explained in these Numbers. Others, placing the two Extreams of the Diatessaron, the acute part in 288, and the lower sound in 216, in all the rest observe the same Proportions, only that they make use of two Remainders of the two middle Intervals. For the Base being forc'd downward

a whole Tone, makes 243, and the upper Note screw'd up a full Tone, begets 256. Moreo∣ver 243 carries a Sesquioctave proportion to 216 and 288 to 256, so that each of the Intervals contains a full Tone, and the residue is that which remains between 243 and 216. Which is not a Semitone but something less. For 288 exceeds 256 by 32, and 243 exceeds 216 by 27, and 256 exceeds 243 by 17. Both which Excesses are less then the half. So 'tis plain that the Dia∣tessaron consists of two Tones and the Residue, not of two Tones and a half. And so let this suffice for the demonstration of these things. Nor is it a difficult thing to believe, by what has been already said, wherefore Plato, after he had as∣serted the differences of Sesquialters, Sesquiterces and Sesquioctaves, when he comes to fill up the Intervals of Sesquiterces with Sesquioctaves, makes not the least mention of Sesquialters; for that the Sesquialter is soon fill'd up, by adding the Sesqui∣terce to the Sesquioctave, or the Sesquioctave to the Sesquiterce.

Having therefore shewn the manner how to fill up the Interval, and to place and dispose the Medieties; had never any Person taken the same Pains before, I should have recommended the further Consideration of it to the Recreation of your Fancies; but in regard that several most excellent Musicians have made it their Business to unfold these Mysteries with a Diligence more then usually exact, more especially Cranter, Cle∣archus, and Theodorus, it shall only suffice to shew how these Men differ'd among themselves. For Theodorus, varying from the other two, and not observing two distinct Files or Rows of Num∣bers, but placing the Duples and Triples in a di∣rect

Line one before another, grounds himself upon that Disposition of the Substance, which is vulgarly call'd the Disposition in Length, making two parts, as it were out of one, not four out of two. Then he says, that the Interpositions of the Mediums ought to take their Places in that manner, to avoid Trouble and Confusion; transferring out of the first Duple into the first Triple the Intervals which are ordained for the supplement of both. But as for those who take Crantor's Part, they so dispose their Numbers, as to place Planes with Planes, Tetragons with Tetra∣gons, Cubes with Cubes, opposite to one another, not taking them in File, but alternatively odd to even. [Here is some great defect in the Original.] Which being in themselves permanently the same, afford the Form and Species; but being subject to Corporeal Division, become the Matter and Subject to receive the others Impression, the com∣mon Mixture being compleated out of both.

Now the Indivisible Substance, which is always one and the same, is not to be thought to be in∣capable of Division, by reason of its Smallness, like the most minute of Bodies, called Atoms. But as it is unmixt, and not to be any way af∣fected, but pure and altogether of one sort, it is said not to consist of Parts, but to be indivisible. By means of which Purity, when it comes in a∣ny manner whatsoever, but to approach and gent∣ly touch compounded, divisible and differing Substances, all Variety ceases and crouds toge∣ther into one Habit by Simpathy and Similitude. But if any one will that Substance which ad∣mits Corporeal Separation, Matter, as a Nature subject to the former, and partaking of it, the Use of that Equivocal Term will nothing disad∣vantage

our Discourse. For they are under a Mistake that believe the Corporeal to be blended with the Indivisible Matter. First, For that Plato does not here make use of any one of its Names, whereas in other Places he calls it the Receptacle and Nurse, capable both to receive and foster the vast Infinity of created Beings; not divisible by Bodies, but ra••her the Body it self, parted and divided into singular Individuals. Then again, what difference would there be, between the Creation of the World and the Soul, if the Com∣position of both proceeded from Matter and preceptible Substances? Certainly Plato himself, as endeavoring to separate the Generation of the Body from that of the Soul, tells us, that the Corporeal part was by God seated and deposited within it, and that it was outwardly covered and inveloped by it: and after he had thus wrought the Soul to its perfection out of Proportion, he then proceeds to this Argument concerning Mat∣ter, of which he had no occasion to make men∣tion before, when he was producing the Soul, as being that which had not its Existence from Mat∣ter. The same may be said against the Follow∣ers of Posidonius. For they seem not altogether to exempt the Soul from Matter; but imagining the Substance of the Extreams to be divisible in reference to Bodies, and intermixing it with the perceptible Substance, defin'd the Soul to be an Idea of some thing distanc'd according to Num∣ber comprehending harmony: seeing that all Mathematick Objects are dispos'd between the first Intelligible, and the first sensible Beings. So that the Soul containing the Sempiternal of things intelligible, and the pathetick Nature of things subjected to Sence, it seems but Rational,

that it should consist of a Substance between both. But they were ignorant, That God, when the Soul was already brought to perfection, after∣wards making use of the Extreams and Limitati∣ons of Bodies to form and shape the Matter, confin'd and environ'd the dissipated and fleeting Substance within the Compass of certain Surfaces compos'd of Triangles adapted together. Nor is it less, if not much more absurd, to make the Soul an Idea. For the Soul is always in motion, the other incapable of Motion; the one never to be mixt with that which is subjected to Sence, the other wrought into the Substance of the Body. Moreover, God could not only be said to imitate an Idea, as his Pattern; but he was the Artificer of the Soul, as of a Work of Perfection. Now that Plato does not assert Number to be the Sub∣stance of the Soul, only that it is order'd and pro∣portioned by Number, enough has been already said.

However this is a common Argument against both the former Opinions, that neither in Corpo∣real Limits, nor in numbers there is the least Foot∣step or appearance of the Power by which the Soul assumes to it self to judge of what is subject to Sence. For it was the Participation of the In∣telligible Principle that endu'd it with Understand∣ing and the perceiving Faculty. But as for Opi∣nion, Belief, Imagination, and its being affected with Qualities relating to the Body, there no Man could ever dream, that they proceeded singly ei∣ther from Unites, or Lines, or Surfaces. For not only the Souls of Mortals have a Power to judge of what is subject to Sence, but the Soul of the World also, so says Plato, when it reverts to it self, and happens once to touch upon any fluid and

roaving Substance; at what time the indivisible part being mov'd by its whole self, gives notice, to what this or that thing, is still the same; to what Heterogenial, to what end, and where, and how it comes to pass that all things act and suffer both upon, and by each other. Soon after mak∣ing a Description of the Ten Predicaments, he gives Us a clearer Manifestation of these things. For true Reason, says he, when it is fix'd upon what is subject to Sence, and the Circle of that other Sub∣stance, mentioned in the beginning, observing a just and equal Motion, conveighs its Intelligence to the whole Soul, then both Opinion and Belief become sted∣fast and certain; on the other side, when it is setled upon Ratiocination; if the Circle of the same Exi∣stence, turning readily and easily, furnishes the same happy Intimations, there Knowledge of necessity arrives to Perfection. And indeed in whomsoever these Accomplishments shall be found, whoever shall affirm them to be the Operations of any thing be∣sides the Soul, may deservedly be thought to speak any thing rather than the Truth.

From whence then does the Soul enjoy this Motion, whereby it recollects by Thought and Apprehension what is subject to Sence, different from that other intelligible Motion, which ends in Knowledge, is a difficult Task to resolve; unless we stedfastly assert, that Plato here did not compose the Soul, so singly consider'd, but the Soul of the World also of the Parts above men∣tion'd, of the more worthy and indivisible Sub∣stance, and of the less worthy divisible in reference to Bodies, which is no other than that Motion which gives Heat and Vigor to Thought and Fancy, and sympathises with what is subject to Fancy, not created, but existing from Eternity

like the other. For Nature, which had the Power of Understanding, had also the Power of Thinking. But the intelligible Power is neither subject to Motion, nor Affection, being esta∣blish'd upon a Substance that is still the same. The other moveable and fleeting, as being en∣gag'd to an unstable, fluctuating and disunited Matter: in regard the sensible Substance was so far from any Order, that it was without Shape, and boundless. So that the Power which is fix'd in this was incapable of producing clear and well grounded Notions, nor any certain or well-or∣der'd Movements, but sleepy Dreams and Deliri∣ums, which amuse and trouble corporeal Stupidi∣ty; unless by accident they lighted upon the more worthy Substance. For it was in the mid∣dle between the Sensible and discerning Faculty, and a Nature conformable and agreeable to both; from the sensible, claiming Substance, and borrow∣ing from Judgment its descerning Power. Which the express Words of Plato declare. For this is my Opinion, saith he, in short, that Being, Place and Procreation, were three distinct things before the Heavens were created. By Place he means Mat∣ter, as being the Seat and Receptacle; by Be∣ing or Existence, the intelligible Nature; and by Generation, the World not being yet created, he only designs that Substance which was subject to Changes and Motions, dispos'd between the forming Cause, and the Thing form'd; trans∣mitting hither those Shapes and Figures which were there contriv'd and moulded. For which reason it was call'd Divisible; there being a Ne∣cessity of distributing Sence to the Sensitive, and Imagination to the Considerative Faculty. For the sensitive Motion being proper to the Soul, di∣rects

it self to that which is outwardly sensible. As for the Understanding, it was fix'd and im∣movable of it self, but being settl'd in the Soul, and becoming its Lord and Governor, whirls about and finishes that Circular Violence which chiefly labours to apply it self to the eter∣nally durable Substance. With great difficulty therefore did they admit a Conjunction, till the Divisible at length intermixing with the Indivisi∣ble, and the restlesly hurry'd with the sleepy and motionless, constrain'd the diversly opposite to be glad of their Society. Yet the diversly Opposite was not Motion, as neither was the Same Stabili∣ty, but the Principal of Distinction and Simili∣tude or Identity. For both the one and the other proceed from a different Principle; the Same from the Unite, the Other from the Duad; and these were first intermix'd with the Soul, being fasten'd and bound together by Number, Proportion, and Harmonical Mediums: So that the Other be∣ing riveted into the Same, begets Diversity and Disagreement; and the Same being fermented in∣to the Other produces Order; and this is apparent from the first Powers of the Soul; which are Judgment and Motion. Motion immediately shews it self in the Heavens, giving us an Ex∣ample of Diversity in Identity by the Circumvo∣lution of the fix'd Stars; and of Identity in Diver∣sity by the Order of the Planets. For in them the Same bears the chiefest sway; in Terrestrial Bodies quite the contrary. Judgment has two Principles; Understanding from the Same, to judge of things in general; and Sence from the Other, to judge of things in particular. Reason is a mixture of Both; Consideration in reference to things intelligible; and Opinion in things sub∣ject

to Sence; making use of the interpos'd Organs of Imagination and Memory. Of which, these in the Same produce the Other, and those in the Other make the Same. For Ʋnderstanding is the Motion of the Considerative Faculty, toward that which is per∣manent and stable. Opinion is a Continuance of that which is perceiv'd by Sence, upon that which is continually in Motion. But as for Fancy or Imagination, being a Connexion of Opinion with Sence, the Same has plac'd it in the Memory: And the Other moves it again in the Difference between Past and Present, touching at the same time upon Diversity and Identity.

But now let us take a Draught of the corre∣sponding Composition of the Soul from the Structure of the Body of the Universe: There we find the pure and limpid Fire, together with the Earth, whose Nature is such as not to admit of Mixture one with another, but with great difficulty; or rather altogether obstinately re∣fractory to Mixture and Consistency. God there∣fore placing in the middle between both, the Air next the Fire, the Water next the Earth, first of all temper'd the middlemost one with another, and next by the assistance of these two, he brought the two Extream Elements not only to mix with the middlemost, but also to a mutual Closure and Conjunction between themselves. Then he drew together the Same and the Other, not immediately the one adjoyning to the other, but placing other Substances between, the Indivisible next the Same, and the Divisible next the Other, disposing each to each in convenient Order, and mixing the Extreams with the Middlemost. After which manner he interweav'd and tissu'd the whole into the Form and Composition of the Soul, com∣pleating,

as far as it was possible, Similitude out of things different and various, and one out of many. Therefore it is alledg'd by some, that Plato erroneously affirm'd the Na∣ture of the Other to be an Enemy to Mix∣ture, as not being only capable to receive it, but a Friend of Change. Whereas that should have been rather said of the Nature of the same, which being Stable and an utter Adversary to Mutabi∣lity is so far from an easie and willing Condescen∣sion to Mixture, that it flies and abhors it, to the end it may preserve it self pure and free from Alteration. But they who make these Objections against Plato, betray their own Ignorance, not understanding that the Same is the Idea of those Things that always continue in the same State and Condition: and that the other is the Idea of those Things which are subject to be variously affected; and that it is the peculiar Nature of the one to disjoyn, and separate into many parts whatever it haapens to lay hold upon; of the other, to cement and assimilate scatter'd and dis∣sentaneous Substances, till they resume one parti∣cular Form and Efficacy. And these are the Pow∣ers and Vertues of the Soul of the Universe. Which when they once enter into the Organs of corruptible Bodies, there the Form of the Binary and boundless Principle shews it self most briskly, while that of the unmixt and purer Principle lies as it were dormant in Obscurity. And thus it happens, that a Man shall rarely observe any such sort of Human Passion or Motion of the Under∣standing, where there shall not something ap∣pear, either of Desire or Emulation, Joy or Grief (which certainly proceed from the more powerful Sway of the Dual Principle in Terrestri∣al

Bodies, as being subject to Disorder and Exor∣bitancy.) Several Philosophers therefore will have the Passions to be so many sorts of Reason∣ings; seeing that both Desire, Grief and Anger are the Effects of Judgment. Others alledge the Vertues themselves to be Passions; Fortitude being subject to Fear, Temperance to Voluptu∣ousness, and Justice to Avarice. Now the Soul being both speculative and practical, contempla∣ting as well Generals as Particulars, and seeming to comprehend the one by the assistance of the Understanding, and the other by the aid of Sence, common Reason, which encounters the Same in the Other, and the Other in the Same, endeavors by certain Limits and Distinctions to separate One from Many, and the Divisible from the In∣divisible: but cannot accomplish her Design, nor be purely in one or the other, in regard the Prin∣ciples are so odly interwoven and intermix'd, and confusedly hudled together.

For this Reason did God constitute a Recepta∣cle for the Same and the Other out of the Indivi∣sible and Divisible Substance to the end there might be Order in Variety. For this was to have a Being, since that without these, the Same cannot be allow'd to have either Variety or Mo∣tion, or Procreation. Nor the Other be said to have either Order or Consistence, or Generation. For should we grant the Same to be different from the Other, and the Other to be the same with it self, such a Commixture would produce nothing Generative, but would want a Third, if I may so call it, Matter, to receive and be dispos'd of by both, and this is that Matter which God first compos'd, when he bounded the moveable Na∣ture of Bodies, by the stedfastness of the Under∣standing.

Now then as Voice, meerly Voice, is only an insignificant and brutish Noise; as Speech is only the Expression of the Mind by significant Utterance; as Harmony consists of Sounds and Intervals; which being mixt toge∣ther produce Air and Melody. Thus the passive Nature of the Soul was without Limits and un∣stable, but afterwards became terminated by that common bound which circumscribes the divisible Variety of Motion, which having compris'd the Same and the Other, by the Similitudes and Dissi∣militudes of Numbers causing Concord of Disa∣greement, becomes the Life of the World, sober and prudent, Harmony it self, and Reason by perswasion overruling Necessity, which by several is call'd Fate or Destiny; by Empedocles Friendship and Discord; by Heraclitus, the opposite straining of the Congruity of the World, like the Strings of a Bow or Harp, whose ends draw several ways; by Parmenides Light and Darkness; by Anaxagoras, Wisdom and Folly; by Zoroastres, God and the Devil, naming one Oromasdes, the other Arimarius. Though as for Euripides, he makes use of the Copulative erroneously for the Disjunctive, where he says,

—Jove, whether he may be Necessity that Natures Force controuls, Or the Intelligence of Human Souls.

For indeed the Powers which bears Dominion over the Universe are Necessity and Wisdom. This is that therefore which the Fabulous Egyptian intimate, feigning that when Orus was punish'd and dismembred; he bequeath'd his Spirit and Blood to his Father, but his Flesh and his Fat to

his Mother; there being no part of the Soul which remain'd pure and unmix'd, or separate from the rest. For that, according to the Opi∣nion of Heraclitus, Harmony Latent, is of greater Value than that which is visible, as being that wherein the blending Deity conceal'd and sunk all Varieties and Dissimilitudes. Nevertheless there appears in the irrational part, a turbulent and boistrous Temerity; in the rational part, an orderly and well marshall'd Prudence; in the sensitive part, the Constraint of Necessity, but in the Understanding, entire and perfect Com∣mand of it self. The limiting and bounding Power sympathizes with the whole and the indi∣vidual, by reason of the nearness of their Relati∣on. On the other side, the dividing Power fixes it self upon Particulars, by virtue of the divisi∣ble Substance: and the whole rejoyces at the Mutation of the Same into the Other, as occasion requires. In like manner, the various Inclinations of Men to Vertue and Vice, to Pleasure and Toyl, as also the Enthusiasms and Raptures of Lovers, the Combats of Honor with lustful De∣sires, plainly demonstrate the Mixture of the Divine and Impassible, with the Mortal and Corporeal Part. Of which Plato himself calls the one Concupiscence of Pleasures natural to our selves; the other an Opinion introduc'd from without aspiring to the chiefest Good. For passi∣ble Qualities of the Souls which are cross'd and hurry'd to and fro by the Affections arise from her self; but she participates of Understanding, as being infus'd from without, by the more worthy Principle, which is God. Nor is the Celestial Nature priviledg'd from this. For sometimes it is seen to encline the other way,

to the more powerful Revolution of the Same.

Nay, there shall come a time, as it has hap∣pen'd already, when the Worlds moving Wisdom shall grow dull and drowzy, drown'd in Oblivion of its own Duty, while that which is familiar, and agreeable to the Body from the beginning draws and winds back the right hand Motion of the Universe, causing the Wheels to go slow and heavy: Yet shall it not be able however to dash in pieces the whole Movement, for that the Better Part rowzing and recollecting her self, and observing the Pattern and Exemplar of the All-directing Deity, betakes her self to speedy Imi∣tation, and thereby retrieves her Negligence, and reduces all things again into their former Or∣der.

Thus it is demonstrable by many Proofs, that the Soul was not the sole Workmanship of the Deity, but that having in her self a certain Por∣tion of innate Evil, it was by him digested and beautify'd, while he confin'd its Infinity to the Ʋnite, to the end it might be a Substance within the Compass of certain Limits; intermixing Order, Mutation and Variety by the Force of the Same, and the Other; and lastly, working into all these, as far as it was possible, a mutual Community and Friendship by the Assistance of Numbers and Harmony. Concerning which things, although you have heard frequent Dis∣courses, and have likewise read several Arguments and Disputes committed to writing upon the same Subjects, it will not be amiss for me also to give a short Account; after a brief Repetition of Plato's own Words. God, saith he, in the first place withdrew one part from the whole; which done,

he took away ths half of that; from thence a Third Part, Sesquialter in proportion to the Second, and Triple to the First: Then a Fourth Part, double to the Second; next a Fifth Part, being the Triple of the Third; then a Sixth, the Eighth Part of the Third; and lastly, a Seventh, being the Twenty Seventh Part of the First. This done, he fill'd up the Duple and Triple Intervals, retrenching also from thence certain other Particles, and placing them in the midst of those Intervals; so that in every Interval there might be two Medieties, the one exceeding and being ex∣ceeded by one and the same part of the Extreams; the other equally exceeding, and being equally exceed∣ed by the same Number. Now in regard that from these Connexions in the first Spaces there arose the Intervals of Sesquialters, Sesquiterces and Octaves, he fill'd up all the Sesquiterces belonging to the Octave Interval, leaving a part of every one, and the distance of the Part so taken from Number to Number, having for their Bounds or Limits 256, and 343. Here the Question will be first concerning the Quantity, next concerning the Order, and in the third place, concerning the Force and Vertue of the Numbers. As to the Quantity, we are to consider which he takes in double Intervals. As to the Order, whither they are to be plac'd in one Row, according to the Direction of Theodorus, or as Cranter will have them, in the Form of a Λamda, placing the Ʋnite at the top, and the Duples and Triples apart by themselves in two several Files. Lastly, we are to examine of what Use and Vertue they are in the Structure and Composition of the Soul. As to the first, we shall relinquish the Opinion of those who affirm, that it is enough, in Proportions, to consider the Nature of the Intervals, and the

Medieties, which fill up their Vacancies: The Demonstration being to be made out of whatsoe∣ver Numbers that have Spaces sufficient to re∣ceive the aforesaid Proportion. For this being granted, it makes the Demonstration obscure, without the help of Schemes, and drives us from another Theory, which carries with it a delight not unbecoming Philosophy.

Beginning therefore from the Ʋ∣nite

	1
2		3
4		9
8		27

let us place the Duples and Tri∣ples apart; and there will be on the one side, 2, 4, 8, on the other, 3, 9, 27. Of which Numbers, including the Ʋnite, two and four make seven, besides that, the Number circumscribing the whole Number, is the Seventh. For not only here, but upon other Occasions, the Sympathy of the Quaternary Number with the Septenary is apparent. For there is this peculiar to that Qua∣ternary Number Thirty six, so much celebrated by the Pythagoreans, for this more particularly wor∣thy Admiration, that it is compos'd of the four first even Numbers; and the four first odd Num∣bers.

The fourth Connexion is made of Numbers put together in order: The first Connexion being of One and Two, the second of Odd. For placing the Unite which is common to both be∣fore, he first takes 8, and then 27, as it were pointing out with the Finger where to place each particular sort.

Even Number.	Odd Number.
1	7
2—2	3—3
4	9
4—8	3—9
32	27
36	36

These Places are so deprav'd in the Original, that the Sense is lost.

But it belongs to others to explain these things more accurately and distinctly, while we content our selves with only what remains, as peculiarly proper to the Subject in hand. For it was not out of Vain-glory, to boast his Skill in the Ma∣thematical Sciences, that Plato inserted in a Trea∣tise of Natural Philosophy this Discourse of Har∣moniacal and Arithmetical Medieties, but believing them both apt and convenient to demonstrate the Structure and Composition of the Soul. For some there are who seek these Proportions, in the swift Motions of the Spheres of the Planets, O∣thers rather in the Distances, others in the Mag∣nitude of the Stars; others more accurate and nice in their Inquiry, seek for the same Proporti∣ons in the Diameters of the Epicycles: as if the Supream Architect, for the Sake of These; had adapted the Soul, divided into seven parts, to the Celestial Bodies. Many also there are, who hither transfer the Inventions of the Pythagoreans, tripling the Distances of Bodies from the Mid∣dle. This is done by placing the Ʋnite next the Fire; Three, next the Earth which is opposite to

our Earth; Nine, next the Earth; 27 next the Moon. Next to Mercury 84. Upon Venus 143, and upon the Sun 729. Which is both a Tetra∣gonal and Cubical Number: from whence it is, that they also call the Sun a Tetragon and a Cube: and by this way of tripling they also reduce the other Stars to Proportion. But these People may be thought to dote, and to wander very much from Reason, if there be any use of Geometri∣cal Demonstration, since by their Mistakes we find that the most probable Proofs proceed from thence; and that though they who most strictly adhere to Probability, do not always make out their Positions so exactly, yet they approach the nearest to Truth, when they say that the Dia∣meter of the Sun, compar'd with the Diameter of the Earth, bears the Proportion of 42 to 1. The Diameter of the Earth to that of the Moon carrys a Tripple Proportion. And for that which appears to be the least of the fix'd Stars, the Diameter of it is no less then the third part of the Diameter of the Earth, and the whole Globe of the Earth to the whole Globe of the Moon is as se∣ven to Twenty One. The Diameters of Venus and the Earth bear a duple, the Globes or Spheres of both an Octave Proportion. The Distance of the Shadow of the Eccliptick to the Diameter of the Moon holds a Triple Proportion and the Deviation of the Moon from the middle of the Signs either to the one or the other side, is a twelfth Part. Her Positions as to the Sun, either in Triangular or Quadrangular distances gives her the Form when she appears as in the first Quarter, and almost at the Full: but when she comes to be quite round, that is, when she has run through half the Signs, she then makes as it

were a kind of Concord of a Diapason. But in regard the Motions of the Sun are slowest when he arrives at the Solstices, and swiftest when he comes to the Equinoxes, by which he takes from the Day, or adds to the Night, the Propor∣tion holds thus. For the first thirty Days after the Winter Solstice, he adds to the Day a sixth part of the Length, wherein the longest Night exceeded the shortest: the next thirty Days, he adds a third Part; to all the rest, till the Equinox, by Sextuple and triple Distances to even the Irre∣gularity of time.

Moreover the Caldeans make a Spring to hold the Proportion of a Diatessaron to Autumn; of a Diapente to the Winter, and of a Diapason to the Summer. But if Euripides rightly divided the Year, where he says,

Six Months the parching Heats of Summer raign; And six of hoary Winters Cold complain: Two Months doth vernal Pride the Fields array, And two Months more to Autumn Tribute pay.

Then the Seasons shall be said to change in Octave Proportion.

Others there are, who fancy the Earth to be in the lowest String of the Harp, according to the most antient Scale call'd Proslambanomenos, or Are, and so proceeding, place the Moon in B mi: Mercury and Venus in C fa ut and D sol re; the Sun they likewise place in Elami, as in the midst of the Diapason a Fifth above the Earth, and a Fourth from the Sphere of the fixed Stars. But neither does this pleasant Conceit of theirs come near the Truth, neither do they in any wise ap∣proach the Accurateness of Proportion.

However, they who will not allow these things to depend upon Plato's Sentiments, yet will they grant the same to partake of Musical Proportions. So that there being five Tetrachords of Base and Tenor, of mean Notes conjoyn'd from Alamire with B flat, and Notes disjoyn'd from B sharp to Elimi sharp, and the Treble Tetrachord from Elimi to Alamire in G sol re ut Clift in these five Di∣stances they place all the Planets; making the first Tetrachord from the Moon to the Sun, all observing the Solar Motion: the next from the Sun to the fiery Planet of Mars: the third between this and Jupiter, the fourth from thence to Saturn, and the fifth from Saturn to the Sphere of the fix'd Stars: So that the Sounds and Notes which bound the five Tetrachords bear the same Proportion with the In∣tervals of the Planets. This might be more pro∣bable among the Antient Musicians, who as well we know confin'd their Scale to seven standing Notes equal in Number to the Number of the Pla∣nets. But the Moders adding the Proslambanomenos or Are, which is a full Tone in descent from B mi, have multiply'd the whole Scheme into the double Diapason, and thereby confounded the Natural Or∣der of the Concords, while the Diapente happens to be before the Tetrachordon, with the Addition of the whole Tone in the Base. Whereas Plato makes his Addition in the upper Part. For in his Politick Discourses, he says, that every one of the Eight Spheres rouls about a Syren, which is fix'd upon each of the tuneful Globes, and that they all sing one unvary'd Counterpoint, and unfigur'd without diversity of Modulation, taking every one their peculiar Concords, which together com∣pleat a melodious Consort.

〈♫〉〈♫〉1 Tetrachord: 2. Tet: 3. Tet. 4. Tet. 5. Tet.

They further add, that their Harmonious plain Song serves them to celebrate several Divine My∣steries no less delightfully useful, while Celestial Voices, according with Heavenly Instruments, may seem to serve as a Recreation to those that are oblig'd continually to dance the Sacred Rounds of Nature. Nor was there Necessity of a fuller Chorus, in regard that within the Confines of eight Notes, lay the first Bounds and Limits of all Duple and Triple Proportions; The Unite being added to the Separations of the Even and Odd Numbers.

And certainly from hence it was, that the An∣tients rais'd their Invention of nine Muses; of which eight were employ'd in Celestial Affairs; the Ninth was to take care of things Terrestrial, and to reduce and reform the Inequality and Con∣fusion of Error and jarring Variance.

Now then consider whether the Soul does not roul and turn and manage the Heavens, and the Celestial Bodies by means of those Harmonious Concords and equal Motions that are wrought and fermented within her; being her self most wise and most just: and such she became by Ver∣tue of Harmonical Proportions. Whose Images are imprinted by the Incorporeal into the dis∣cernable and visible Parts and Bodies of the World. But the Chief and most predominating Power is mix'd in the Soul, which renders her obsequient and obedient to the most supream and divinest Part of all the rest at the same time, u∣nanimously consenting. For the Soveraign Arti∣ficer and Creator finding a strange Disorder and erroneous Confusion in the Motions of the discom∣pos'd and unruly Soul, which was still at variance with her self, some things he divided and separa∣ted,

others he brought together, and reconcil'd to a mutual Sympathy, making use of Harmony and Numbers. By Vertue of which the slightest and meanest of insensible Substances, even Sticks and Stones, the Roots of Plants, the Rinds of Trees, the Skins and Excrescences of Beasts, with the Superfluities of their Concoction, according to their various Mixtures, Compositions and Tem∣peratures, some become the charming Objects of the Sight, others afford most pleasing perfumes and wholsome Medicaments for the Succour and Relief of Mankind, while others are wrought and hollow'd to send forth those pleasing Sounds that ravish even the Soul it self with the Re∣flections of her conceal'd Beauties and Concinni∣ties. And for this Reason it was that Zeno the Citizen encouraged and perswaded Youth to fre∣quent the Theaters, there to observe the Variety of melodious Sounds that proceeded from Horns or Cornets, wooden Haut-boys, Flutes and Reeds, or any other Musical Instruments, to which the Contrivance of Art had rightly apply'd the Rea∣son of Number and Proportion. Not that we will here maintain with the Pythagoreans that all things resemble Number, for that requires a long Discourse to prove it. But where mutual Socie∣ty and Sympathy arises from Discord and Dissimi∣litude, that the Cause of this is Moderation and Order, was a thing not conceal'd from the less studious Poets; who therefore to Persons full of Humanity, sweet of Disposition, and friendly, gave the Epithite of evenly concinnated: On the other side, Men of rugged and malicious Dispo∣sitions they call'd Ʋnevenly Temper'd, as if Enmi∣ty and Discord were nothing but a sort of Dis∣proportion. For this reason, he who writes

Pindarus's Elegy gives him this Encomium.

—So were his Manners fram'd That Strangers still his sweet Demeanor fam'd; To all Domestick born a Friend so true, That they that knew him only Friendship knew.

The Poets plainly hence inferring Complacen∣cy of Humor, and the Aptitude of a Person to fit himself to all Tempers to be an Excellency aspiring to Vertue it self. Which Pindarus him∣self also testifies speaking of Alcimedon.

He fears not Orcus nor the Stygian Night, Who acts in Consonance with Truth and Right.

Nor must we believe that the Theologists, who were the most antient Philosophers, order'd the Pictures and Statues of the Gods to be made with musical Instruments in their Hands, as if they thought the Gods no better than Pipers or Harpers, but to signifie that nothing so much de∣noted the Structure of the World to be the Ma∣ster-piece of a God, as the Order and Sympathy of the Creation.

Now then as it would be absurd and ridicu∣lous for any Man to search for Sesquiterces, Ses∣quialters and Duples in the Neck or Belly or Sides of a Lute or Harp (though every one of these must also be allow'd their Symmetry of Length and Thickness) the Harmony and Proportion of Concords being to be sought for in the Sounds; so 'tis most probable that the Bodies of the Stars, the Distances of the Spheres, and the Swiftness of their Motions and Revolutions, as instrumental Organs, have their sundry Proportions as well one

to another, as to the whole Fabrick, though the Measure of the Quantity be unknown to us. How∣ever we are to imagine that the Principle, Effect and Efficacy of these Numbers and Proportions, which the supream Architect made use of, is that same Agreement, Harmony and Consent of the Soul with it self, by means of which Numbers she replenish'd the Heavens themselves, when she came to actuate and perform her Office there, with so many infinite Beauties; and governs the Earth by vertue of the several Seasons, and other Alterations wisely and artificially measur'd and vary'd as to Mixture and Temperature, as well for the Generation as Preservation of all Ter∣restrial Productions.