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dianged at least twice and recommended by the com- the new cycle of Epacte in harmony with the year Yxy

mission in a third or even later form. two equations so called, the solar and the lunar. The

The opposition of the Sienese Teofilus against the solar equation diminishes the enacts by a unit when-

innovation of the Epacts was supported by Alexander ever a Julian leap day is omitted, as in 1900; the lunar

Piccolomini, coadjutor Bishop oi Siena. If he was not equation increases the epacts by unity every 300

a member of the commission, he was at least re- years, or (after seven repetitions, the eighth time) in

quested to express an opinion. He laid down his 400 years. The former eauation accounts for the er-

tneories in a " Libellus on the new form of the ecclcsias- ror in the Julian year and the latter for the error in the

Tables he gave preference to Albategni's length of the that greater exactness could be reached by applying

year and advocated the correction of the Julian inter- the interval of 400 years the tenth time. It may hap-

calation once in every hundred years (thinking the pen that the two equations cancel each other and leave

error to amount to one day in 106 years). Piccolo- the epacts unchanged, as happened in 1800. The new

rius, Clavius, Ciaconus, Lilius, Dantes, all mentioned Kings of France, Spain, Portugal, from the Dukes of

above. The last mentioned, usually called Ignazio Ferrara, Mantua, Savoy, Tuscany, Urbino, from the

Danti, was afterwards made Bishop of Alatri. His Republics of Venice and Genoa, from the Universities

scientific reputation may l)e inferred from the praises or Academies of Paris, Vienna, Salamanca, Alcali,

given to him more than a himdred years later (1703) Cologne, Lou vain, from several bishops and a number

by Clement XI for his large solar instruments in of mathematicians.

Rome, Florence, and Bologna, which affirmed the cor- The Bull "Inter Gravissimas '*. — ^The contents of the rectness of the Gregorian equinox. The instruments answers are not officially recorded, but in the Bull of consisted of meridian lines and gnomons. The former Gregory they are called concordant. How the concord- were usually strips of white marble inset in stone ance is to be understood may be illustrated by the floors. The gnomon was sometimes replaced by a answers from Paris and from Florence. While the small opening in a wall, which projccte<l the image of Sorbonne not only rejected the "Compendium" but the sun on the meridian line. An arrangement of this condemned every change in the calencuir, the king's description is visible in the old Vatican Observatory, Parlement fully adopted the reform proposed Dy called the Tower of the Winds. It was on this line Lilius. The Duke of Tuscany forwardea to the pope that, according to Gilii and Calandrelli, the error of ten the judgments of several Florentine mathematicians, days was demonstrated in the presence of Gregory no two of which agreed among themselves, while he XIII. himself gave full approval to the Gregorian reform.

The manuscript of Lilius was never printed and has The King of Portugal presented two professional never been discovered. Its contents are known only answers without adding a judgment of his own. The from the manuscript report of the commission and emperor also confined himself to forwarding the reply from the "Compendium" of Ciaconus, which was from the University of Vienna. The answers from printed by Clavius. The request of Clarante, that his Savoy, Hungary, and Spain were in approbation of " Calendarium " be distributed together with the Lilius's plan. All the princes may have seen the ne- " Compendium '*, was not granted by the commission, cessity of a reform and desired it. This is confirmed The "Compendium" was sent out in 1577 to all Chris- by a letter of the Cardinal Secretary of State to tian princes and renowned universities, to invite ap- Charles Borromeo, Archbishop of Milan, dated 16 probation or criticism. With Lilius, it left open the June, 1582, in which the statement is made that the questions, whether the equinox should be placed on 24 reform of the calendar was concluded witi tiie appro- March or 21 March, following the old Roman Calendar bation of all Catholic princes. The consent of the or the Council of Nicspa; and if the latter (which princes had more influence with the pope than the seemed preferable), whether the ten days should be opinion of scientists. To bring about an agreement of omitted at once, in some suitable month of 1582, or the latter was utterly hopeless, and, in view of the gradually by declaring all of the next forty years com- labours of the papal commission, unnecessary. The mon years and thus completing the reform in 1620. variety of opinions, collected by Kaltenbrunner and That the error from the Nica^an regulation of the equi- Schmid, bears testimony to this, quite apart from the nox had amounted to ten days, was sufficiently known bitter polemics that followed the Gregorian reform from various observers, like Toscanelli, Danti, Coper- and which does not concern us in this article, nicus (Calandrelli, "Opuscoli Astronomici ", Rome, The propositions made in answer to the "Compen- 1822, 30). The motions of sun and moon were taken dium" may be summed up as follows. In regard to from the Alphonsine Tables. \Vhether the Prutenic the solar year, the date ot the equinox should be 25 Tables of 1554 were at the time known to Lilius may March, where Julius Ca?sar had put it — this was the be doubted. He could be no stranger, however, to wish of the Humanists — or 24 March, where it was at Cardinal d'Ailly's " Exhortatio ad Concilium Con- the time of Christ's resurrection — this was the propo- stantiense", in which the Julian intercalation was sal of Salamanca — or 21 March, where the Coimcil of showTi to be one day in error everv 134 years, or to the Nicaea had put it, or finally should be left on 1 1 March, proposition of the Veronese matbematician Pitatus, where it was at the time. Those who would not ac- who wanted the correction applied by a cycle of four cept the correction of the Julian intercalation by full centuries. Lilius considered fractions of centuries un- centuries wanted a leap day omitted as often as the fit for all cyclic or non-astronomical reckoning and error amounted to a full day — ^by the Alphonsine used centurial corrections for both solar and lunar Tables every 1'34 years —, or, as the theological fac- motions. ulty of the Sorbonne demanded, no correction at all.

Lilius's masterpiece is the new "Nineteen Years' As to the lunar cycle, no university attempted an im-

C>'cle of Epacts ", by which he kept the Nica?an Easter pro vement on Lilius's epacts. Salamanca and Alcali,

regulation apace with the astronomical moon. The asweknowfromalctterofCla\'ius to Moleto in Padua,

old lunar cycle gave the lunations four or more clays fuUv approved Lilius's reform. Vienna rejected all

in error, and Easter could thus (by taking the Sunday cvclical computation, whi4c the theolo^cal faculty of

after Luna XIV) fall on Luna XXVI, within a few the Sorbonne pleaded for the retention of the old

days of the astronomical new moon. liilius brought cycle, uncorrected. The ans^^'ers from Louvaixi de-