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antiqui ty. We are told, for instance, that certain paris of the Scriptures are written not in Hebrew, but in Chaldaean. The preci Se relations of these two languages ought to he made familiarto US 73 776. Sixthly, the Latin texi of the Scriptures, RS Common ly u Sed, is extrem ely Corrupi, and beComes more So aS time goeS On. It need sCaresul correction by referetice to the original Greeli and Hebreis. The versions in recent use by the Franci Scans and Domini Cans aresar inferior to the old versionS . . 77-817. Seventhly, When the texi is CorreCt, there is osten the greatest Obscuri ty as to the interpretation. The Same Latin word Correspondsosten to many tota, ly distinct Hebre ords 81 858. Eighthly, and lastly, since Latin grammar is formed on themodet os Greek and Hebrew grammar, muCh Confusion ariSeS Domitiis Sotirce. Latin worus os fore ign origin are not reCOgni red; anu ConverSely, to many wOres of Latin Origin a fore ign source is
diphthongs in Greela that corresponcl to a Latin vo wel. The rules for gender that hold good in Latin are not applicabie to Greeli. The fame holdS good os pronunciation. The penultimate os poSAeSSive adjectives, whicli is long in the Case of words of Latin origin, uS bovinum, is Short where they are derived froni Greela, as in Suchword S RS Crystallinum, adamantinum 85-92
In Conclusion, I must pOint out the importance to the Church of linguistic studies ; si in for explanation of the liturgy, 2 of the formulaeu Sed in sacraments and consecration S, 3ὶ for the due regulation offorei gn Churches, 43 for throwing light on the future history of the Church, s) for interco urSe Willi foretgn nations . . . . 92 96
I pasS now to mathematics, the Mundation and the key to allother scien CeS, Studi ed frona the earli est ages of the worid, but os late fallen into neglect. I shali deal successively with iis application to human knowledge, to divine knowledge, and to the g Overn ment Osthe Church 97 98
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There is high authority for this estimate os mathemati s. In the study os divine things and also of man's sociat life Boethius and Ptolemy show that it is of great service. The various modes of proportion have their analogue in Civit poli ty. It is needed both in Daminar and logi C, as Alpharabius and Cassiodorus have sito n. For prosody dependS enti rely on arithmetical relations. Logic, oniis practical fide, haS the Same purpose as poetic and rhetoric, whi Chdepend ora harmony. Further, the subject-mat ter os logic is intimately Connected with mathematic. This is obvious in the Categories of quantity, of time, and of place. In the Category os quali ty much helongs to the mathematical domnin, RS O g. geometrical form. The Same may be said of the Category of relation. Spiritual substances Can only be known through the medium os corporat: and the firSt Step to the knowledge of body is the studyof the heavenly bo dies. The dependenCe of astronomy on mathemati S is obvious 98 1 03CHAPTER III.
We reach the Same Conclusion by reasoning. sa) In ali other sciences we uSe mathematical examples, beCause they illustrate thepoint without consuSing complications ; e. g. in explaining the differen e
who easi ly appreciate the simple arithmetical relations on whichmusic dependS. gi We acquire our knowledge of things known toourseives more east ly than of things known to nature si . e. intringi catly Simpler as, for instance, the trullis of theology). But mathemati s have the double character of being both relatively and absolutelysimpler. hi In mathemati s demonstration i S more Complete: iis Cogeia Cy has the force of necessi ty, whicli is not the Caseeither in physics, in metaphysic, in et hic. a) In other sciences the uncertainty of the premisses involves uncertainty in the Conclusion. These principies require verification by Some Science more perfeci than them selves: i. e. by mathematics. lj Finalty, the subject-matter of mathemati Cs is more directly Cognigable by oursenses. It deals with quantity, which lies at the root of ali knowledge. The Simplest process of intellect implies continuous quantity,
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i inpossibi e to malae progress without mathemati S. The number and motions of the heavens, planeta motion S, eclipSeS obviouSty requiret his science. And things terrestriat no leSS, SinCe they are governed by things celestiat. The sun, acting on the Surrounding medium,
Rays passing Dom a rarer medium to a denser, is they impinge u ponthe lalter perpendicularly, purSue a reCtilinear CourSe. Otherwisethey are refracted, i. e. diverted to arcis the perpendicular drawn to thesuriace of Contact. In passing Dona denser to rarer they are diverteuaway Dom this perpendicular. Heiace is the denser body be spherical, the Solar rays passing through it Converge at a gi ven poliat beyond it. The Convergence of many rays produces heat. Is the second medium is so dense that the species' Cannot pasS through, or at te St that iis transit Cannot be appreciated by human vision, it is reflected. Isthe ray falis perpendicular to this medium, reflexion is in the reverse direction os inciden e. Is otherwise, the angle os reflexion is equat tot he angle of inciden Ce. By Concave reflectors Solar rnyS may be Concentraled. Is the reflector is spherical, tho se rays will be focussed whicli impinge on potnis in it corresponding to a circle placed atright angi es with iis axis. But reflectors Can be devi sed of Such formiliat ali the rays shali fati on the reflecting sartaCe at equat angleS, and thus be reflected to the fame focus. We have further to Considerthe diffusion os light impinging on objecis not direct ly froin the Sunbut in direct ly. Constant exposui e to direct rays Would be destruCtive. Lastly, we have to consider the effect of light on the nerves of vision, when there is no question os iis following any rectilinear CourSct,
iis path being modi fied by the vitai principi e . . . 111 - 117
Rays is suing in infinite number Dom a pol ni in every directionfind their termination on hollow surtace of a Sphere. Each Potnt of
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civ ANALYSIS OF THE OPUS MAIUS
the sursace a ted on is the vertex of a cone os rays. of whicli the baseis the whole sui face of the agent 117 119 TM1RD DISTINCTIO N.
Light and other sorces not meret y propagate themgelves by multiplication os species, but work ulterior effecis; light produces heat, heat putrefaction, putrefaCtion deuth, and So on. TO these effecis the fame law applies. Rectilinear action is more effective than curvilinear; perpendicular than oblique. In refraction the effect is greater than in reflexion, be ause in the lalter the reflecte land incident rays neutralige each other. In refraction action is stronger Where the second medium is denser than the first, beCausethe ray is deflected towards, not away frona, the perpendicular. Inreflexion more is done by oblique rays than by perpendicular. Withthe lalter there is neutraligation os incident and reflected rays. Further, there can be but one perpendicular ray, but infinite numbersos oblique ; and these may ali he made to Converge . . . 119 123
The rays of whicli a natural action Consist forna, RS we have Seen, a Cone. In the Shorter Cones the Strength of the action is promoted
first by greater proximi ty of the agent, Secondi y by the greater
proximity of the Conterminal rays after intersection. On the other hand in the longer Cones, the ra S before interSection are the nearerto each other, and in this respect the aCtion will be more potent. But the first of these conditions will outweigh the second . . 123 124
When two equat spheres interaci, the half of each whicli is averted froni the other i S unaffected ; the extreme rayS DOm eaCh Can on lyembrace the half of the Other sphere. But with unequat spheres, thel ess receives rays frona less than the hemisphere of the greater, whichtouch more than iis own hemisphere. Froni each potnt of a sphererays issue into space Out si de of the tangent plane. of these rayson ly one is perpendicular to the furface. Thi S i S the potent ray. RayS VertiCal to a Sphere are divergent. But when the object ofvi Sion iS very remote, as in the Case of stellar hodies, they appearto us to be parallel: just as the walis of a house Seem parallel, although their lines of direction converge to the earlli 's centre . 124 127
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By the foregoing principies and others ah in to them, whicli forivant of Space are here omitted, ali natural actions are to be explained. A seis illustrations may be given. The planetS reCei vetheir light froin the sun. Hence when the earth is belween the sunand moon, the moon is eclipsed by the earth's Shado A. The coneos this sita dow not reaching however to the Other planet S, theSe arenot eclipsed. Again, that the eye and the starS Can mutuatly transmittheir emanations through the media of the planetary OrbitS, ProVe Sthese media to be of great rari ty, to be invi Sible, and non-luminouS. The sphere of fire also is nei ther luminous nor visibi e . A planet differs froni the sphere in whicli it moves by greater condensation of elestiat substance; hence iis luminosi ty. This, though ultimatelyderived frona the sun , is not due to the sun 'S reflected rays. The diffusion of moonlight proves the inoon to have independent luminosi ty. Owing to the magnitude of the sun being Iro times greater than that of the moon, much more than half of the earili is illuminated by the sun ; and the Same is the case with the moon and planets. Other assections of the moon's light depend on her varyingconjunctions with planeis and with con Stellations . . . 127-I30
The Same principies may be applied to disprove the allegeo simplici ty of cosmic Structure. A Star on the meridian is seen to befurther from the pote than at iis rising. In the lalter position refraction dis places it. This stlows the worid to consist os distinctSubStan CeS of va ing densi ty ; for a ray would not be refracted while pasSing through the Same substance, even though iis paris si, ouid beos different densi ty . . . 130-132
Hence it is that the temperatures of the variouS Zones are eXplained. Beneath the potes the cones of rays are prolonged, and thereire feeble : Capable only of rai Sing vapours Dorn the earth and sea, Sothat the air of those Climates is heavy and cold, and un fit for livingthingS. Neverthele SS. owing to the tength of the days and of the twilight in summer, the sun being ne ver sar distant, there may be places in those regions, favoured by the position and inclination os Certain m Ountains, where the rays are So reflected that the climate is
tem PCriate ........... 132-l35
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Passing to the torti d Zone, it Would Seem that the region under theoquinoctial circle must be the liottest, since twice in the year the sun svays there are Verti Cal. But this is over-weighed by the faci that under the tropical signf the Sun i S nearly stationary. The matter is further Complicated by the e Centrici ty of the Solar orbi t. . 135 137 CHAPTER U. The emanations froin the starS affect not meret y climate, but Character ; implanting on the new-born child dispositions to goodor evit, to qui k or to duli apprehension : though Dee will, God's grace, temptations Of the devit, Or education may modisy these innate
tendenCiOS ........... 137-139
Cur theory may be applied to the lides. These evidently dependon the moon. When the rnys fali obliquet y on the sursace, their effect is only to rat se vapoiars hom the Suriace and create ebullitionand a consequent fow of water illi the time comeS when the rays fallverti Cally, and with sorce enough to extraci the vapour; and then there flux begins. This however leaves it unexpiat ned why the samething happens in the hemisphere averted frona the mOOn. V e muSt Suppose the ninth or starry heaven to be solio and impenetrabie, and that the vertical rays of the moon are reflected hom it, these producing the fame effect as the incident rays .... 139-142
The application of these principies to the preservation of life and health is obvious. Protection must be fought against the vertical ra, S of injurious emanation S, RS of the moon at night, of Saturn and Mars, of persons in Cted With contagious disense, of the evit eye; and we must adapt our bodies to the reception os emanations knownto be SalubriouS . . . . . . . .. 142-143
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God. Nor is it enough to say that matter is infinite potentiat ly, butnot in esSence. Nor that it is potentialty infinite in the sense in whicli this is snid os continuous quantity. For to attribute to matterexistence in indefinite numbers of substances is to attribute to itin fini ty, not meret y potentially, hut in act. The contradiction in whichthis lands us may be set fortii geometri catly. Nothing infinite Canhave finite power, and conversely nothing finite Can have infinite PoWer 143 148CHAPTER IX.
When two spheres are brought together, and the stra ight lines Domtheir Centres to the poliat os Contaci are ContinuouS, the questionarises whether these lines beCome one, or Whether We are toregard them as two. Averroes maintained a distinction belween mathematical quantity and natural quantity. But this distinctionis unienable. The lines in question are two, although they have the effect of one, and for convenienCe of speech may be spolien ofaS one. Against the separabili ty of disterent masses of matter itis argued that is two circular planes are brought into Contactand then separated, air wili penetrate into the o uter portion besore the in ner, hence sor a moment there will be a vacuum in the Centralpari. But the angwer is that the Separation is not simultaneou Sthroughout the whole suriace of the plane, So that the air penetratOS gradually. Frona the divisibili ty of matter, it is not to be argued that the worid is composed os an infinite number of material parti cles, asLeucippus and Democritus maintained. Were this so, it might bein serred that the diameter of the square was Commen Surable with iis
On geometrical grounos the Shape of the universe Caia be inferreuto be Spheri Cal. No other forin would preclude the possibili ty ofa Vacuum in the Course of it S revolution. Cylindrical or lenticular form Would sum Ce is revolution took place round a Certain axis. Withthe spherical forin revolution round whatSoever axis would avo id vacuum. Looked at Dom within, it must be Concave and spherical: Other i se lines di a n froni the Centre of the earth to the extremittes of the universe would not be equa l. Further, the Sphere is that form hi Ch under a gi ven Suriace has the greatest content. It is the Simplest and nobi est os formS. The water, the air, and the sire Surround ing the ear th concentri Cally, are of similar forni . . 152-lbi
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Suppose two vesseis similar in s hape and equat in Sige ; one placedat a higher level than the Other. More water can be placed in thelower, sor iis furface will be a portion of a Amalter sphere: thediameter of the rim of the vesset be ing equat in both cases . 157 159
The Platonic school maintained that heaven and the Dur elements correspondeu to the live regular SolidS. For there Can be no morethan five. Since in the docle hahedron the other Mur can be inscribed, thi S was regarded as representing heaven fre was identi fied withthe tetrahedron, air with the optahedron, earth with the Cube, water V ith the icosi hedron. But the dissiculty in this theory is that, though sol id masses can be bulli up of tetrahedra and of Cubes with Out leaving vacua, thi S is not the case with the other three . 159 161
Unity of time cloes not imply uni ty of matter. Nor is it needful to Suppose plurali ty of ages saeva). The subject of time is not matter, but motion. The subject of motion is not matter, hut body composed of matter and form. Motion is of linear dimension. Prior excludes POSterior, past eXClude S future. But as to the present being a potnthaving no dimension, there is no Such exclusion ; One potnt cloes notexclude another : many potnis occupy the position of one. OnepreSent moment sumces for ali preSent momentS. HenCe time is one. And So to the Conception of aevum the Same applieS. It is Single and
not multi Ple .......... 165-167
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When two equat weights are placed in the scales of a balance, andone scale is depressed, the lower Scale being nearer the earth'S Centre. will have greater gravi ty. But this is counterbalanced by the tenden cy of the upper scale to fati in a Curve os more rapid descent than the lower. The oscillation of the arms of the balance above and below the horigontal is due to the motion communicated to the Surrounding air . 169 174
THE APPLICATION OF MATHEMATICS TO SACRUD
We have now Seen the potera cy of mathemati s as applied to thingsSeCular. V e now pass to iis application to things divine. Philosophyis impossibi e without mathematics: theology Without philosophy. All knowledge is contained, directly Or indirectly, in Scripture. Therefore for the right understanding of Scripture, knowledge of nature is needful. In Scripture there is a doubie mean ing, litera land spiritual. The first is neces Sary for the Secono and , aS we ha VeShown, mathematical knowledge is necessary for it. It is certain that the patriarchs studi ed mathematical science and transmitted it to the Chaldaeans and Egyptians, whence it came to the Greelis. This is proved by Josephus, and confirmed by Jerome and other doctors, andalso by Such philosophers as Album agar. Further, the falliers havethenaseives extolled the value os mathematical science, a S may beShown by passages Dona Cassi OdoruS, Augustine. Bede and otherS. The importance of mathematics to theology may be Considered under Seven head S 175 180 First head. Knowledge of the heavens. Astronomy Sho Sthe insignificance of the earth as comparect with the heavens. Thesmallest of the stars is larger than the earth. and the largest Star is insignificant compared with the space of the sky. The earlli Canbe traversed at tot paCe in three years. A star moving with immenSe Veloci ty talaes thirty-si x thousand years to CompaSS the henVen S. Further, it is to astronomy that we look for solution of many theological problems, as for the substance of whicli the heaven S are made, the position os paradi Se and of heli, the influence of heavenly bOdies tapon the things of earlli. Again many obscurities in the texi ofSCripture Can only be Cleared up by astronomical research . 180-183 The second head is that os sa Cred geography. By geogra Phy, which is dependent on astronomy, we Can determine the Preci Seposition and the physical conditions of the places named in Scripture.
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All these, apari stom their literat importance, have a distinci spiritualsignification. The river symboliges the worid ; the D ad Sea, heli ;Jericho, the flesti ; the Mount of Olives, spirituat illa; the valley of Je hostia phat, humili ty ; Jerusalem, the foui in the eMoyment of
peace, Or again the Church militant and triumphant. MinutereSearch will reveat numberless intermediate mean ings . . 183 187 The third head relates to sacred chronology. SCripture preSentSto us a Succe SSion Of times, with regard to whicli precise knowledge Can only be gi ven by mathematical astronomy. The Stari ing-pointis the creation of the worid. Was this in the autumnal or the vernalequi nox Θ From what i S said in the old Testament as to the Feastos Ingathering Exodus xxiii. 16J we should inser the former. VetJewish and Christian commentators adopi the lalter view. It wili besor astronomy to decide this difficuli potnt. Further, the question of the longevi ty of the ancient patriarchs has to be Considered. Onemode of accounting for it may be the more favourable position of the sun and the planeis in primitive times. Again there is the problemof the Deluge. The right interpretation of Josephus potnis to November as the monili in whicli it hegan. Lastly, dici night comebeire day, or the Converse Θ The former would seem to be the true Vie 187 195The murth head deals not with chron olom in generat, but withthe definition os periods. How is the beginning of a lunationto be fixed y by astronomicat calculation, or by the moment when thenow moon is visibi e Z The actuat lunation is variable. The average lunation must be uSed. The Jews use the Metonie Cycle of n ineleenyearS, or 235 lunation S. This gives t enly-nino dayS, t etve liourS. and of an hour for the mean lunation. They take a period of thirteen lunar Cycle S Or 247 years, within whicli ali their festivals recur atthe Same moment. The lunation is considered to begin with the.sun- set immediately following the Computed time. These considerations
may be applied to the clate of the Creation, of Noah's issue froni thearli, of the Passover, and finalty of the Passion. The current belles in the Latin Church is that Christ was horn in the second year ofa lunar Cycle. and di ed on Marcii 23 a. d. viii Kal. Aprilis), the moonbeing at the fifteenth. day the Greelis holding that it was the Durteenth dayj. Against this much may be urged. It impli es thatit was in the thirteenth year of a Cycle. Frona the Computation OfS. Dionysius this would involve the PasSion tali ing place on a Sunday, whicli is impossibie. A table is appended showing one solution of the difficulty. This would show the date of the Passion to be April 3 a. d. iii Non. Aprilis), on the fifteenth year of the lunar cycle, Christbeing then thirty-two years old. This vlew is offered to the Pope forcon Sideration 195 210