Since I will focus on the Ratio Studiorum and technology,
let me begin by situating the Ratio in the context of the
young Jesuit Order, and also by situating “technology” in the
context of the disciplines mentioned in the Ratio.
Ignatius of Loyola gave the Society of Jesus and the world two
major documents: The Spiritual Exercises and The
Constitutions of the Society of
Jesus.1
The Exercises
focuses on an individual and on his or her personal
relationship with God and others. In contrast, the
Constitutions focus on a group—the Society of
Jesus—and on its members, but also includes thoughts on a key
Jesuit work: education, normally a group experience.
After Ignatius's death in 1556, it was felt that both of these
documents needed commentaries to explain his thoughts and to
offer additional guidance. As a result, 400 years
ago,2 two
other documents appeared. On January 8, 1599, the definitive
version of the Ratio Studiorum was approved, expanding on
Part IV of the Constitutions, and, on October 1,
1599, the definitive version of the Directory on the
Spiritual Exercises.
Part IV of the Constitutions actually refers to a
more detailed, “separate treatise” regarding
schools,3 and the
Ratio was that document.4 Even though neither the Constitutions
nor the Ratio could mention any technology available to us
today, in particular my field of computer science, nevertheless
both refer to the foundational disciplines needed for any
technology, namely, physics and mathematics. We should also note
that the “mathematical sciences” in the sixteenth century
included applied topics now considered separately as part of
astronomy, physics, and engineering.5 Thus, I suggest, both the
Constitutions and the Ratio can continue to give
guidance as we stand on the brink of a new millenium and reflect
on Jesuit Education and on new academic and technical
disciplines.
It is important to situate the Ratio among the significant
events in the history of science. About 55 years before
the 1599 Ratio appeared, Nicholas Copernicus published
De revolutionibus orbium clestium;6 about 17 years
before the Ratio,7 Pope Gregory XIII ordered the shift to the Gregorian Calendar based on the
astronomical and mathematical work done by Jesuit Father
Christopher Clavius; about 35 years after the Ratio,
Galileo was condemned;8 and about 90 years
after, Sir Isaac Newton published his Philosophiæ
Naturalis Principia Mathematica.9 Thus the Ratio
appeared about the same time as the beginnings of the
scientific revolution, and so it is not surprising that the
references to modern science it contains may seem sparse.
Another major factor to consider is the lack of prestige given in
the sixteenth century to mathematics and to what we now call
scientific and technical disciplines. At that time, a number of
Italian philosophers, including some Jesuits, denied to pure
mathematics the status of scientia, true scientific
knowledge in Aristotle's sense, because it did not demonstrate
its conclusions through causes and it dealt with abstractions in
the intellect, rather than real objects.10 Thus, mathematics was
not considered, by many, to be worthy of study in a university.
Mathematics was presented via the subjects of the
quadrivium, namely arithmetic, geometry, music (taught as
applied arithmetic), and astronomy (taught as applied geometry),
and also, in some degree, via logic (one subject of the
trivium). But, at that time, the trivium and
quadrivium were considered preparatory arts studied before one
pursued the higher disciplines of theology, medicine, or law,
which were usually the only accepted disciplines for university
study, particularly in Italy.11
Evidence of the hostility toward mathematics in that era is
provided in the 1591 draft of the Ratio, which includes a
curious admonition12
directed toward university administrators to make sure that
philosophy teachers do not disparage the dignity of mathematics.
This rule concludes with the astute observation, “for it often
happens that the less one knows about such things, the more he
devalues them.”13
Christopher Clavius was professor of mathematics at the Roman
College for about 45 years, from 1565 until his death in 1612.
Although not officially a member of any commission involved with
the Ratio,14 Clavius's influence on the
Ratio's sections on mathematics cannot be denied.
Clavius saw the value of mathematics and did not feel himself
bound by the Aristotelian categories that gave it a relatively
low place in the hierarchy of academic disciplines. In contrast,
Clavius wrote: “Since ... the mathematical disciplines in
fact require, delight in, and honor truth ... there can be no
doubt that they must be conceded the first place among all the
other sciences.”15 This
predates Gauss's statement that “mathematics is the queen of the
sciences” by about two centuries!
In actuality, three different versions of a Ratio were
issued by the Jesuit Curia: two preliminary drafts in 1586 and in
1591, and the definitive version of 1599. There also exists a
slightly revised version of the 1586 text that was never
published.16
Around 1580, Clavius authored two documents, probably written for
the commission charged with composing the first, 1586
Ratio.17 Similarities between the
sentiments expressed in these documents and in the 1586 and 1591
draft Rationes along with the inclusion of Clavius by name
in these drafts attest to their influence.
The first document, Modus quo disciplinæ mathematicæ
in scholis Societatis possint promoveri,18 blamed philosophy instructors, in part, for the low opinion
held by many about mathematics since some “teach that
mathematical sciences are not sciences.”19 Clavius bemoaned the poverty of mathematical
interest and instruction, and argued that the lack of
well-trained mathematics teachers and the absence of Jesuits
studying mathematics harmed the Society.20 The document also notes that without mathematics one
cannot understand most physical phenomena.21
Clavius's other brief note, De re mathematica
instructio,22 dealt with
faculty problems and the need to have well-trained, mature
instructors.23 Its sentiments
influenced the 1586 Ratio which prescribes at least
two mathematics instructors at the Roman College, naming Fr. Clavius as a possibility for one of these positions.
The impact Clavius had on the 1586 Ratio cannot be
underestimated. He was someone who had done significant
scientific and mathematical work, his work in revising the Julian
Calendar being only one example.24 And he proclaimed the importance of
mathematics in the face of an academic culture in Italy and
professorial colleagues at the Roman College who apparently
demeaned mathematics whenever possible. Clavius saw new uses for
mathematics in ways that were only beginning to be discovered in
the last decades of the sixteenth century. He wanted to make
sure that Jesuit schools placed proper emphasis on mathematics,
by recruiting and training qualified instructors of mathematics
as well as by requiring that mathematics be studied by all.
Clavius's daring proposals are now viewed as revolutionary. He
did not want partial remedies but a complete and complex strategy
to obtain a radical change in attitude and
structures.25
Due to Clavius's influence, the 1586 Ratio included a
lengthy apologia about mathematics and its connection to
various disciplines and professions, including poetry, history,
politics, metaphysics, theology, law, farming, and medicine. It
requires teachers of mathematics in gymnasia and two
professors of mathematics at the Roman College. All students are
required to take mathematics for at least a year and a half and
an academy of eight to ten Jesuit students would be established
to study specialized mathematics.26 After their studies, these
young Jesuits would be able to travel to various countries and
use their mathematical knowledge wherever they
went.27 Although this last
comment may seem odd, we should recall that the Jesuit
mathematician Matteo Ricci had studied under Clavius in Rome
about ten years earlier28 and in 1586 was actually
using his mathematical acumen to gained credibility in China,
initially by translating two of Clavius's mathematical books into
Chinese. Thus, at the same time mathematics was demeaned by
philosphers in Italy, mathematics and astronomical technology
became the doorway by which Ricci was able to gain entry into the
culture of China.29
Some modern scholars view the 1586 Ratio as a radical
document for raising mathematics to be on a par with other
university-level disciplines and for giving it a prominence
unheard of in Italian universities.30 I suggest that Clavius's insight about
the importance of the mathematical sciences, incarnated in the
1586 Ratio, is a model for how we approach teaching
technology in Jesuit schools in the next century. Unfortunately,
the culture and the times of the sixteenth century were not
appropriately receptive to this insight.
The advice contained in the 1586 first draft of the
Ratio31 seemed too radical for some people. This draft was
submitted by a Spanish Jesuit, Father Enrique Enriquez, to the
Spanish Inquisition. His contention was that the Ratio was
at variance from the teachings of St. Thomas Aquinas, much of
whose outlook was Aristotelian.32 Changes were introduced into the
Ratio, and some of the differences between the 1586 draft and
the definitive 1599 Ratio are noteworthy in light of what
scholars have recently written about these changes.33
One striking difference pertains to the amount of text devoted to
mathematics. In the 1599 Ratio, the references to
mathematics are only about a quarter of the length of what
is found in the first 1586 draft.34
The never published 1586 second draft kept the apologia for
mathematics found in the first draft, but revised the practical
advice. This new draft added a public presentation of
mathematical problems before an assembly of students once or
twice a month, and monthly repetitions of the principal topics in
an interactive format. In contrast, the number of students
mentioned for special tutoring is reduced from eight or ten to
only four or five. Perhaps the revisors of the Ratio felt
that Father Clavius was too optimistic in gauging the number of
Jesuits competent for studying advanced mathematics.
The 1591 draft did not include any rationales why various
disciplines were to be taught. So, the general apologia
for mathematics disappeared. In addition, the study of
mathematics by all is reduced to one year. A second year is only
recommended for students of metaphysics, and the special
academy is reduced from three years to six months.35
The definitive 1599 Ratio makes no direct mention of how
long students should study mathematics, but only says that
mathematics should be taught to the students of physics and those
in the second year of philosophy. It also reduces the public
presentations of mathematical problems from once or twice a month
to each month or every other month.
When the Constitutions list the subjects taught in
universities, it says, “Logic, physics, metaphysics, and moral
philosophy will be treated, and also mathematics, in so far as
they are in accord with the end proposed to us.”36 The Latin reads “et etiam
mathematicæ.” The presence of the etiam seems to
give an added emphasis, as if one would not normally expect
mathematics to be taught at a university.
The general norm found in the Constitutions about what
should be taught in universities is this:
Since the end of the
learning which is acquired in this Society is with God's favor to
help the souls of its own members and those of their neighbors,
it is by this norm that the decision will be made, ..., as to
what subjects [Jesuits] ought to learn, ....37
Elsewhere, we find this advice:
since the arts or natural sciences ... are useful for the
perfect understanding and use of [theology], and also by their
own nature help toward the same ends, they should be treated with
fitting diligence ....38
The thrust of Ignatius's thought seems obvious: Any
discipline, including scientific, and therefore technological
disciplines, which “help[s] souls” and “by their own nature help
toward the ... ends” envisioned in the Constitutions,
is worthy of study in a Jesuit university.39
Early Jesuit education was innovative in introducing into
universities the study of mathematics alongside philosophy and
physics.40 Especially in the first 1586
Ratio, mathematics was presented as a key to understanding
physical reality as well as the model of correct rational
procedure.41 But, as already noted, merely prescribing
that mathematics be taught in a Jesuit university did not
eliminate all the anti-mathematical prejudices of scholars from
other disciplines.42
Since mathematics is an abstraction, it does not directly solve
societal or spiritual problems. But, mathematics trains the mind
to think in disciplined, structured ways, offers insights into
the diversity of creation, and provides the language for the
world of science and technology. Following Galileo, one can even
consider that the “book” that is the universe “is written in
the language of mathematics.”43 Thus,
in its own way, it contributes to the common good and to
improving our global society.
One remarkable aspect of both the Constitutions and the
Spiritual Exercises is their flexibility. Time and again,
we note references to adaptation according to individuals,
places, and cultures.44
This is also true of the Ratio and the Directory.
And one adaptation that schools must continually make is to
modify the subjects taught as well as the methods used to teach
newer subjects. Unfortunately, even with explicit
recommendations to adapt as needed, in actuality Jesuits were
unable to accommodate the Ratio to the changes in the
world, especially in the last two centuries. The attempt in 1832
to issue a revised Ratio ended in disaster since its
contents were obsolete and it seemed to go against the currents
of educational history.45
Unlike the debates about mathematics in the sixteenth century,
today there is little disagreement about the appropriateness of
disciplines such as computer science or engineering being offered
in a university or about their academic stature. Centuries ago,
the Church seemed to be an enemy of science and technology, with
the Galileo condemnation in the seventeenth century and the more
recent prohibition of railroads and gas lights in the papal
states by Gregory XVI in the early nineteenth
century.46 But in the twentieth century, the Church has
taken a significantly different approach.
Perhaps it was John XXIII who introduced the new emphasis
when he wrote:
the progress of science and the inventions of
technology show above all the infinite greatness of God ...
And since our present age is one of outstanding scientific and
technical progress and excellence, one will not be able to
... work effectively from within unless he is scientifically
competent, technically capable and skilled in the practice of his
own profession.47
The recent Apostolic Constitution, Ex Corde Ecclesiæ,
speaks to the work of Catholic universities in the fields of
science and technology.48 The current existence of a Vatican web site and of Catholic
programs on television and radio is a welcome sign that the
Church has embraced technology to help spread the gospel message.
The Ignatian tradition of “finding God in all things,” even in
technology, is accepted without debate today. Thus, perhaps the
more important questions deal with the opportunities and
challenges that studying technological disciplines presents to
Jesuit schools and their students.
The renowned anthropologist Margaret Mead once said: “The
solution of adult problems tomorrow depends upon the way we raise
our children today.”49 We
cannot deny that the future leaders of corporations and nations
are those students in school today. And in a world in which
science and technology are commonplace, leadership in tomorrow's
world demands familiarity with technology and with the ways it
helps or harms human progress.
Today's technology provides opportunities never before possible
in our world. The early Jesuits saw education as a way to
transform society, and the Ratio provided structures to
provide an excellent education. In a sense, including
mathematics in the Ratio forced Jesuits and their students
to become familiar with this foundational language used by
intellectuals of a world on the brink of what we now call “the
scientific revolution.” Through mathematics, Jesuits and their
students became engaged, perhaps unknowingly and unwillingly,
with a culture that was only beginning to make its influence
felt, the culture of Galileo and Newton, among others. Today,
Jesuits and their students can again become engaged with a new
culture, the culture of technology and the Internet, through
knowledge of computers and technology.
The accessibility of modern technology enables Jesuit education
today to continue the tradition of transforming society and
engaging culture in new and exciting ways. In North America,
Jesuit universities are finalizing plans for a common
distance learning program, in which students do not have to be
present physically at a university to take a course. There are
already several accredited American universities that are
completely on-line. Who knows how education will change as
technology develops? Such developments offer the Society of
Jesus opportunities for sponsoring new modes of education as well
as a challenge to do it well. It is entirely possible that
in 10 years there will be several virtual Jesuit
universities, all run from computers housed in closets perhaps at
the Jesuit Curia in Rome, or at a community in Manila or
Kraków.50
There is also a great benefit in Jesuits' using their computer
expertise to promote spiritual growth. Creighton University's
web site on spirituality has had a tremendous impact since its
creation in 1998 as well as the Irish Jesuits' Sacred Space
web site. The Society could sponsor an academic electronic
journal on the Spiritual Exercises, accessible to anyone on
the planet—much cheaper and reaching a wider audience than
paper versions, and keeping with the Ignatian principle of having
an impact on more people.51 We
must also not limit our vision based on today's technology, much
of which has been around for only about ten years and may be
hopelessly obsolete in ten more years. What the technology of
100 years from now will enable Jesuits to do is anyone's
guess!52
I suggest that the Ratio provides insights that can still
be applied when making decisions about new disciplines,
especially technology, in contemporary Jesuit schools.
One insight deals with the need to study new disciplines, even if
their stature is controversial. The vision of Clavius found in
the preliminary Rationes was that mathematics was so
necessary that everyone should study it for at least a year and a
half, and that some should study it in greater depth. I suggest
that a similar regulation is desirable for technology
today.53 The spirit of the Ratio
demands that all students enrolled in Jesuit schools become
familiar with technology as a self-contained
discipline54 as well as learning how to use
technology as a tool for information and communication, since
such knowledge brings them into intimate contact with today's
culture.
A related insight is that Jesuits themselves need to be trained
in new technical fields. To my knowledge, I am one of only three
American Jesuits with doctorates in computer science. The early
Rationes went against academic culture in recommending that
Jesuits learn mathematics. Perhaps we must do something similar
today and do it quickly, for some have already suggested that in
the use of technology and the web, we are doing “too little, too
late.” It is a tribute to the early Society that over 30 lunar
craters are named after Jesuit scientists.55 Can we regain that tradition in the next
century?
Another insight deals with public presentations of mathematical
topics. The Ratio recommended regular lectures, perhaps
similar to a “Colloquium Series” at American universities, in
which mathematical topics are presented, even by
students,56 with those present questioning the presenter, even
suggesting alternative proofs.57 The early Rationes
suggested showing how mathematics is useful in the arts or in
daily lives, and similar topics could also be included in
lectures about new technologies.58 I recommend that such lectures on technological
topics become a regular feature in our schools.
A fourth insight deals with the creation of a private
“academy,” that is, a study group for more capable
students.59
Particularly in technical and scientific fields, which many
students find difficult, smaller groups of students, special
seminars, or Honors classes can be taught aspects of an academic
discipline inaccessible to the general student population.
A final insight is the need to respect the integrity of every
discipline. Although the Rationes stressed the pervasive
usefulness of mathematics, they also ordered that no one should
speak disparagingly about a subject unknown to him. I suggest
that this insight means that Jesuit schools should be places
where basic scientific and technological research is honored and
encouraged, even if the field does not seem immediately
practical. It is contrary to the Society's tradition of the
magis to teach or become involved in sciences and new
technologies only half-heartedly!
The Ratio also provides us with challenges as well.
Scientific and technical expertise divorced from a vision of a
better world is hollow knowledge. We must never fear scientific
progress, but we must also never let science become our masters
and we its slaves! Technology, like the Church, is a reality
which is semper reformanda. That on-going change puts an
added demand on those who teach and attempt to learn technology
in ways most other academic disciplines do not.
There are ethical and societal questions raised by technology for
which, even now, the world seeks answers, and Jesuit schools
provide privileged places for experts in ethics, theology,
science, and technology to share their thoughts on the what
and how as well as the why and what for. We
also must address questions about a new elite and a new poor that
are raised by the access and lack of access to technology and the
information it provides. There is also the danger of students'
allowing technology to think for them, rather than understanding
technology to be able to improve it and our world.
I wonder what Ignatius and Clavius would have done if they had
had the opportunity of educating and evangelizing society
electronically? That may be pointless speculation, but it does
provide the seeds for our dreams about the future of Jesuit
education.
The purpose of education has developed over the centuries. In
Roman times, education was seen to develop pietas, a
reverential devotion to parents, the gods, and society. There
was a moral aspect to learning and an implied duty to use
knowledge to benefit society, following the thought of Cicero,
who said “we are not born for ourselves alone.”60
After the scientific revolution, the goal of education in many
places shifted to discovering veritas, which is the one
word motto of Harvard University. Truth itself is the
goal, without any sort of connection to the self or to society.
I detect another stage that is becoming prominent today,
namely, that education is utilitas, in the sense of being
something useful and profitable for the self. I am
actually surprised if I do not hear students ask questions about
what type of job they can get if they choose to study a certain
discipline.
It is easy for many students to become enamored of today's useful
technology because it can lead to prestigious and well-paying
jobs. Although I cannot eliminate this as providing some
students with incentive for studying, I suggest that Jesuit
schools also need to safeguard their ambience lest power and
prestige seem to be the sole reason why Jesuits carry on their
educational apostolates. The reason we teach and search for
veritas is to develop in ourselves and our students
pietas.
The Constitutions remind us that “the end of learning
which is acquired in this Society is ... to help the souls
of its own members and those of their neighbors.”61 Whatever helps build up God's
Kingdom on earth, whatever helps ensure that God's justice will
prevail, whatever helps promote peace and unity among peoples and
relieves pain, disease and poverty, whether it involves the
classics or the latest modern technology, that is what “helping
souls” is about, and that is what Jesuit education in the next
millenium is about as well.
Context of the Ratio within the Society of
Jesus
Context of the Ratio and the World of
Knowledge
The Vision of Christopher Clavius
Changes between the 1586 Ratio and the 1599
Ratio
Role of Mathematics and Technology
Modern Technology and Contemporary Jesuit Higher
Education
The Purpose of Higher Education
1 Both of these documents express the saint's thoughts about human beings and their relationship to God and to other people, but each document in a distinctively different way. Cf. Schineller. Full bibliographic information about books and articles cited is found in the reference section at the end.
2 That is, 59 years after the papal approbation of the Society of Jesus and 43 years after Ignatius's death.
3 Part IV, Chapter 13, [455].
4 The Ratio may also be seen as a type of “directory,” that is, a collection of “directions” expanding on the relatively brief notes in the Constitutions about subjects to be taught and the pedagogical methods to be used in Jesuit universities. Just as the Directory for the Spiritual Exercises emphasizes that one must take into account the state of the retreatant and adapt the retreat accordingly, I suggest that we must take inspiration from the original Ratio, adapting its insights to a changed body of knowledge available for study in the contemporary world.
5 See the introduction by Edward C. Phillips, S.J. to the translations of Documents 34 and 35 (of Clavius): “For a better understanding of some portions of these documents, it should be remembered that at that period ‘Mathematics’ was a term including astronomy and much of what would now be taught in physics.” Also, cf. Homann, p. 81, “... mathematics courses comprised not only arithmetic, geometry, and algebra, but also diverse use of mensuration and calculus in astronomy and astrology, computation of time (calendar and sundial), surveying, theory of music, optics (perspective), and mechanics.”
6 On the Revolutions of the Heavenly Spheres (1543)
7 In 1582.
8 In 1633.
9 Mathematical Principles of Natural Philosophy (1687)
10 Dear, pp. 36-37; Homann, p. 5; Wallace, p. 136.
11 Homann, p. 81; cf. Dear, p. 35. The arts and humanities were only beginning to make their way into the universities, to be followed by the sciences as we think of them today.
12 Probably inspired by Clavius.
13 “fit enim sæpe, ut qui minus ista novit, his magis detrahat.” Ratio Studiorum, 1591, Regulæ Præpositi Provincialis: de mathematicis, n. 44; cf. Homann, p. 79, endnote 69. A translation of the sections pertaining to mathematics in all the Rationes is found in Appendix A.
14 In an article by Gabriel Codina, S.J., Clavius's name does not appear among the members of any commission charged with writing or revising various drafts of the Ratio. Cf. Codina, pp. 8-9.
15 Dear, p. 38, In disciplinas mathematicas prolegomena, in Opera mathematica, Vol 1, p. 5. Cum ... disciplinæ Mathematicæ veritatem adeo expetant, adament, excolantque, ..., quin eis primus locus inter alias scientias omnes sit concedendus.
16 The second 1586 version was recently found in the Jesuit archives, cf. Homann, p. 34. Neither of the 1586 documents were first drafts; both had had several predecessors, the best known of which was the Ratio of St. Francis Borgia, compiled between 1565 and 1572.
17 Homann, pp. 61, 64.
18 “A Method by which the mathematical disciplines can be promoted in the schools of the Society,” This is Document 34 in the 1901 Monumenta Pædagogica S.I. A translation is included in Appendix B.
19 Homann, pp. 61-62; Dear, p. 35; Clavius, “A Method ...,” (Doc. 34), Latin text, p. 473, præceptores philosophiæ ... in quibus docent scientias mathematicas non esse scientias.
20 cf. Homann, p. 61-64.
21 Homann, p. 62; Clavius, “A Method ...,” (Doc. 34), Latin text, p. 472, cum tamen apud peritos constet physicam sine illis recte percipi non posse, .... Some of the thrust of this apologia for mathematics seems to have influenced the content of the 1586 Ratio if not the exact wording as well.
22 “Concerning mathematical instruction,” This is Document 35 in the 1901 Monumenta Pædagogica S.I. A translation is included in Appendix B.
23 cf. Homann, p. 63.
24 The details of the calendar computations are found in his treatise, Novi calendarii Romani apologia, (A Defense of the New Roman Calendar) which appeared in 1588 and was over 800 pages long. We should also note some of Clavius's other contributions to mathematics, such as the use of x as an indeterminate unknown quantity (Cajori, § 161, p. 154; MacDonnell, p. 29), the decimal point (Cajori, § 280, p. 322; MacDonnell, p. 18), parentheses in algebraic expressions (Cajori, § 161, p. 151, § 351, p. 392; MacDonnell, App. I, p. 5). He seems to be one of the first in Italy to make use of the square root sign (Cajori, § 327, p. 369).
25 Homann, p. 64; Feldhay, p. 221.
26 The text states that the second professor, who could only be Fr. Clavius, would teach students in the special academy.
27 Cf. Feldhay, pp. 221-22.
28 Ricci studied at the Roman College from about 1572 until 1577.
29 Immersion into and dialogue with culture is a contemporary Jesuit theme as well, cf. General Congregation 34, Decree 4, “Our Mission and Culture.”
30 Feldhay, pp. 221-22; Dear, p. 35.
31 Regarding both mathematics and other disciplines.
32 Farrell, p. 231; Feldhay, p. 222.
33 cf. Feldhay, p. 223ff. Of course, since “technology” does not appear in the Ratio, the changes relevant to our discussion deal with mathematics.
34 In the first 1586 version, the section dealing with mathematics runs to about 54 lines in a recent edition of the Latin text. In the 1599 version, the section dealing with mathematics is only 10 lines in the same edition, with four additional lines about studying mathematics in a different section.
35 The philosophy curriculum at the Roman College in the late 16th century consisted of a three-year cycle, with logic taught the first year, natural philosophy and mathematics the second, and metaphysics the third. Cf. Wallace, pp. 6-8, 59-61.
36 Part IV, Chapter 12, [451].
37 Part IV, Chapter 5, [351]
38 Part IV, Chapter 12, [450].
39 Smolarski, p. 109-10.
40 Dear, p. 35.
41 Feldhay, p. 222. Thus mathematics serves the understanding of the physical world as well as of ultimate, i.e., metaphysical reality.
42 Cf. 1591 Ratio, rule 44 concerning mathematics of the Rules for Provincials, cited earlier.
43 Galileo, The Assayer (1623), cf. Drake, pp. 237-38. The pervasive nature of mathematics was emphasized in the 1586 Ratio influenced by Clavius whom Galileo knew. Cf. Wallace, pp. 91, 269.
44 E.g., ConstSJ [455], “... [the separate treatise on schools] ought to be adapted to places, times, and persons, ...”; ConstSJ [477], “... that which is found to be more suitable may be done.” ConstSJ [526], “These details and those which follow below are appropriate and should be observed when possible, but they are not necessary.” SpirExerc [18], “The Spiritual Exercises must be adapted to the condition of the one who is to engage in them, that is, to his age, education, and talent.”
45 Codina, p. 15; Donohue, p. 53.
46 cf. Kühner, Encyclopedia of the Papacy, p. 225
47 Pacem in Terris, Encyclical Letter, April 11, 1963, n. 3, 148.
48 August 15, 1990, Par. 7, 18, 45.
49 Quotation found from web pages of quotes without reference to any published works of Mead.
50 Such universities could have excellent courses offered by an outstanding faculty selected from every physical Jesuit university and accessible to anyone with a computer, no matter where they live on our planet or even orbiting our planet. On the other hand, we should not ignore the possible negative aspects that can occur in some individuals who spend long hours before computers. cf. Carnegie-Mellon University, HomeNet Project, 1995, http://homenet.andrew.cmu.edu. But the benefits might be worth the risk.
51 ConsSJ [622-623]
52 The ongoing development of human knowledge along with the changes that such developments introduce into our human society at the least challenge us to rethink established practices. As a young Jesuit, I had to ask permission of the provincial superior to bring a manual typewriter to philosophy studies since my typewriter had a few special mathematical characters on the keyboard. Having a personal typewriter had been seen as a variance from the normal rules governing the practices of poverty. Today I have two computers in my room in the Jesuit community and two others in my office. The perennial challenge is to follow the Jesuit tradition of being intimately involved with the culture in which we live and work, yet not let it dilute any of our core traditions of Christian and consecrated life.
53 It was only in 1995 that my own university approved a new requirement of at least one technology course for all students, even though personal computers had been on campus for ten years and the school is located in the middle of what is called “Silicon Valley.”
54 This may even mean that everyone be required to write a computer program.
55 MacDonnell, pp. 74a, 74b, 76.
56 1599 Ratio, Rule 2 for the Professor of Mathematics
57 1591 Ratio, Rule 3 for the Professor of Mathematics.
58 1591 Ratio, Rule 3 for the Professor of Mathematics; 1586B Ratio, De mathematicis.
59 Although the early Rationes made specific mention of an academy for mathematics, the same concept can be used for other scientific and technical fields as well.
60 Marcus T. Cicero, de Officiis, Book I, vii.
61 Part IV, Chapter 5, [351]
[1] The Constitutions (4th Part, Chpt 12, Par C [451]) say, “Logic, physics, metaphysics, and moral philosophy will be treated, and even mathematics, in so far as they are in accord with the end proposed to us.”
Nevertheless, they seem to be in accord not a little, not only because without mathematics our academies would be lacking a great ornament, indeed they would even be mutilated, since there is almost no moderately celebrated academy, in which the mathematical disciplines do not have their own (and indeed the highest) place; but even much more, because the other sciences also very much need the help of mathematics. For indeed, the mathematical disciplines supply and explain to the poets the rising and setting of heavenly bodies, to historians the shape and distance of places; to those who analyze (analyticis) examples of solid proofs (demonstrationes); to political leaders easily admired techniques to manage both domestic and military affairs well; to physicists the forms and distinctions of heavenly revolutions, of light, of colors, of transparent media, of sounds; to metaphysicians the number of spheres and intelligences; to theologians the major parts of the divine handiwork; to law and ecclesiastical custom the accurate reckoning of time. For the meantime, we pass over the utilities which flow to the republic from the labor of mathematicians in the healing of diseases, in voyages by sea, in the pursuit of farmers. Therefore, an effort must be made so that, just like the other disciplines (facultates), mathematics also may flourish in our schools (gymnasiis), so that from this also Ours will become more suited for serving the various interests of the Church; especially since it is not a little unseemly that we lack professors who are capable of having a lecture about mathematical topics, longed for in so many, such famous, cities. At Rome also, if you subtract one or perhaps a second person, there will be hardly any one of those left who is qualified either to instruct about these disciplines, or to be at hand for the Apostolic See, when there is a discussion about ecclesiastical times.
2. So that we may remedy such scarcity and want, we require two mathematics professors in the Roman College. Let one of these prepare a short course (curriculum) of mathematical topics for a year and a half with daily lectures to be heard by Ours and by externs; the professor should begin this course after the Pasch of the Resurrection in the morning at first school hour for students of logic; because at this time generally they are preparing themselves for the Posterior Analytics, which without mathematical examples can scarely be understood; and since they may be at that time a bit more advanced, they seem not unequal to the burden of three lectures. Nevertheless, it is appropriate that things be arranged that some of the thornier Elements of Euclid be always seasoned by an interpretation either of geography or of the sphere; since in particular these topics do not greatly need knowledge of all the principles of Euclid, but of certain of the first [principles], which after around two months will have been explained. After this, of the three-quarters of an hour to which a mathematical lecture is limited, to the two earlier [topics], the sphere or other more acceptable topics of that sort should be handed over, and this arrangement should be persevered even to the end of the studies. Afterwards, during the second year, the remaining part of the compendium of mathematics, to be completed by Fr. Clavius, will be expounded to the same students, who then will be studying physics, in the first hour of the classes after the midday meal. When, however, Easter approaches, let there be added for the benefit of new students of logic a second early morning lecture, in which the compendium of mathematics is begun again. This presentation and repetition in the same order is to be observed each year.
3. Let a second professor, who could only be Fr. Clavius, be appointed; let him provide fuller teaching about mathematical topics over three years, and teach privately about eight or ten of Ours, who are at least of average ability and not adverse to mathematics, and have studied philosophy; these should be recruited from various provinces, one from each one, if it is possible. And there will be not a few who will be eager to be of the number of these [special students], if, after philosophy in the time others are teaching humane letters, they devote themselves to mathematics; then also to theology; in this way, surely, as in the first two years, let them study nothing other than mathematics. But in the third year, [they attend] also two lectures of scholastic theology with a brief repetition of them, which will take place only in the schools; but the entire remainder of the day they give themselves to mathematics. Afterwards, the excellent mathematicians of this academy will go forth, who will disseminate this discipline into all provinces to which they will return, and they will uphold the reputation of Ours, if at any time it behooves them to speak about mathematics. And it will not be a problem for any province to devote some one of their philosophers also for a third year to mathematics, with the hope of such excellent fruit.
[1] The Constitutions (4th Part, Chpt 12, Par C [451]) say, “There will be treated logic, physics, metaphysics, moral philosophy, and even mathematics, in so far as they are in accord with the end proposed to us.”
Nevertheless, ... when there is a discussion about ecclesiastical times. (as in the first 1586 draft.)
Therefore, let the professor of mathematics explain to all in physics in the school after the mid-day meal for about three-quarters of an hour the Elements of Euclid; in these [explanations], after they have become somewhat familiar during two months, let him so divide the time for the lecture that he devotes some to Euclid, some to geography or to the sphere or to other things which are wont to be heard gladly.
Once or twice a month, let some one of the students explain in detail some famous mathematical problem before a large gathering of philosphers and theologians, after first having been taught it thoroughly by the teacher, as necessary.
Furthermore, on one Saturday of each month, in place of the lecture, let the principal points that during that month had been explained be publicly repeated, not in an uninterrupted speech, but with the students mutually asking questions of themselves, generally in this manner: Repeat that proposition. How is it proven? Can it be proven otherwise? What use does it have in the arts or in the other practices of common life? For these things also should be pointed out by the teacher during his lecture, by which he can attract students the more.
From these philosphers, who will have completed the repetition of philosophy in six months, for another period of the same semester of the year, let there be at home, either through the same or through another professor, an academy of mathematical topics, which for this use Fr. Clavius will have collected in a certain compendium. Let there be a lecture twice a day, and let the progress be as full as possible. And let the students not become involved with any other studies, but let them give themselves entirely to listening to, repeating, and discussing mathematics.
In addition to this private academy, Fr. Clavius would gladly put together his work in another private academy likewise of four or five of Ours, to whom, if they are already philosophers, and are at least of average ability, and not hostile to mathematics, and are free from other studies for two years only, he will explain the primary parts of the mathematical discipline, with the hope of excellent fruit.
It seems both obvious and in our interest not to endure, that so great an opportunity will slip away from us just as from our hands, but rather that several men be nourished who, as successors to Clavius and to his teaching just as heirs, will disseminate it into various provinces, wherever they are sent; and they will uphold the dignity of the Church as well as the reputation of Ours, if at any time it behooves them to speak about mathematics.
This then, if it is to be performed, it should be performed as quickly as possible; both because if any difficulty of this work opposes itself now, it will not be less in the future; therefore he who is not prepared today, will be less so tomorrow; and because uncertain is the end and boundary of life; and while we have a human being so distinguished in this science, it seems the task should not be dragged out to chance outcomes.
1. Let him explain to all in physics in the school after the mid-day meal for about three-quarters of an hour the Elements of Euclid. In these [explanations], after they have become somewhat familiar during two months, let him so divide the time for the lecture that some is devoted to Euclid, some to geography or to the sphere or to other things which are wont to be heard gladly.
2. Once or twice a month let one of the students explain in detail some famous mathematical problem before a large gathering of philosophers and theologians, after first having been taught it thoroughly by the teacher, as necessary.
3. Furthermore, on one Saturday of each month, in place of the lecture, let the principal points that during that month had been explained be publicly repeated, not in an uninterrupted speech, but with the students mutually asking questions of themselves; generally in this manner: Repeat this proposition. — How is it proven? — Can it be proven otherwise? — What use does it have in the arts and in the other practices of common life? — For these things also should be pointed out by the teacher during his lecture, by which he can attract students the more.
4. Where it is possible to be done conveniently, either the same professor at different hours, or at the same hour two professors should have two daily public lectures, in which they explain some of mathematical curriculum to be written by Fr. Clavius, for two years, indeed in the first year to the physicists, and in the latter year to the metaphysicians; although in this latter year, Ours should neither be compelled nor admitted, unless the superiors grant permission to those seeking it.
5. In addition to these public lectures, from the same philosophers of Ours, those who will have completed the repetition of philosophy and ethics in six months, for another period of the same semester of the year let there be at home an academy of mathematical topics, which for this use Fr. Clavius will have collected in a certain compendium. Let there be a lecture twice a day, and let the progress be as full as possible. And let the students not become involved with any other studies, but let them give themselves entirely to listening to, repeating, and discussing mathematics.
41. In another period of the same semester of the year from the same philosophers let there be at home an academy of mathematical topics, which some industrious and well skilled professor twice daily will explain to Ours; they are strictly forbidden not to be involved at that time in any other studies, but to give themselves entirely to listening to, repeating, and discussing mathematics. Let there be in these topics fuller progress to the extent possible according to the compendium of Fr. Clavius; and those who particularly excel and are not of a spirit alien from this subject, let them be set apart for this study as much by frequently expanding on it in the private academies, as by speaking about it publicly when there will be an occasion for it.
42. Let all the philosophers in the school during the second year of philosophy after the mid-day meal for about three-quarters of an hour listen to a mathematical lecture from the Elements of Euclid; in these [lectures], after they have become somewhat familiar during two months, let the time for the lecture be so divided, that some is devoted to Euclid, some to geography or to the sphere or to other things which are wont to be heard gladly.
43. Where it is possible to be done advantageously, either the same professor at different hours, or at the same hour two professors should have two daily public lectures, in which they explain some of mathematical curriculum to be written by Fr. Clavius, for two years, indeed in the first year to the physicists, and in the latter year to the metaphysicians; although in this latter year, Ours should neither be compelled nor admitted, unless the superiors grant permission to those seeking it.
44. Let those who preside beware most strictly, that the professors of philosophy not disparage the dignity of mathematics while teaching or at other times, and not rebut their [mathematical] sentences, as about epicycles; for it often happens, that the less one knows about such things, the more he detracts about them.
1. What Authors Are to Be Explained, and at What Time and to What Students. — Let him explain in class to the students of physics for about three-quarters of an hour the Elements of Euclid. In these [explanations], after they have become somewhat familiar during two months, let him add something of geography or of the sphere or other matters which are wont to be heard gladly, and this along with Euclid either on the same day or on alternate days.
2. Problems.—And let him arrange that every month or every other month some one of the students before a large gathering of students of philosophy and theology has some famous mathematical problem to work out and afterwards, if it seems well, to defend his solution.
3. Repetition.—Once a month and generally on Saturday in place of the lecture, let the principal points that have been explained during that month be publicly repeated.
20. Students and the Time for Mathematics.—In the second year of philosophy all the students of philosophy shall attend class in mathematics for three-quarters of an hour. If there are some, moreover, who are fitted and inclined towards these studies, let them be practiced in them in private lessons after the end of the course.
NOTE: Translation of the 1586, 1586B, and 1591 Rationes by Dennis C. Smolarski, S.J. Sections of the 1586 Ratio compared with the partial translation found in A. C. Crombie, “Mathematics and Platonism in the Sixteenth-Century Italian University and In Jesuit Educational Policy,” PPISMATA (Prismata): Naturwissenschaftsgeschichtliche Studien (Festschrift für Willy Hartner), pp 62-94 (particularly pp 66-67) and with the translation found in F. A. Homan, S.J., Church, Culture and Curriculum: Theology and Mathematics in the Jesuit Ratio Studiorum, passim (particularly, p 78). Translation of the 1599 Ratio revised by Dennis C. Smolarski, S.J. and based on that by A. R. Ball, in Edward A. Fitzpatrick (ed.), St. Ignatius and the Ratio Studiorum, New York: McGraw-Hill Book Co., Inc., 1933. Thanks is due Prof. Helen Moritz, Department of Classics, Santa Clara University, for advice.
In the first place a teacher (magister) must be chosen with uncommon erudition and influence (authoritate); for if either of these be lacking, the students (discipuli), as experience teaches, seem unable to be attracted to the mathematical disciplines. Now in order that the teacher should have greater influence over his students and the mathematical disciplines themselves be of greater value and the students may understand their usefulness and necessity, the [mathematics] teacher should be invited to the more solemn events (acta) at which doctorates are conferred and public disputations held, so that, if he be capable, he may also propose arguments and assist the disputants. For in this way the students, seeing the professor of the mathematical arts attending, with the other instructors, these sorts of events and sometimes even disputing, will be convinced that philosophy and the mathematical sciences are connected, as they truly are; especially since up to now the students seem almost to have despised these sciences for the simple reason that they think that they are not considered of value and are even useless, since the person who teaches them is never summoned to public events with the other professors.
It also seems necessary that the instructor (præceptor) should have a certain inclination and propensity for lecturing on these [mathematical] sciences, and should not be taken up with many other occupations; otherwise he will scarcely be able to help his students. Now in order that the Society may be able always to have capable professors of these [mathematical] sciences, some men apt and capable for undertaking this task ought to be chosen who may be instructed in a private academy in various mathematical topics; otherwise it does not seem possible that these studies will last long in the Society, let alone be promoted; since, however, they are a great ornament to the Society, and quite frequently a discussion about them will occur in conversations and meetings of leading men, where they might understand that Ours are not ignorant of mathematical topics. Whence it happens that in such meetings Ours necessarily become silent, not without great shame and disgrace; as those to whom this very thing has happened have often reported. I omit mentioning that natural philosophy without the mathematical disciplines is lame and imperfect, as we shall show a little later.
This much has been said about the teacher of mathematical disciplines; now let us add a few words on the students (auditores).
In the second place, therefore, it is necessary that the students (discipuli) should understand that these [mathematical] sciences are useful and necessary for the correct understanding of the rest of philosophy, and at the same time that they are as a great ornament to all other arts, so that one may acquire eruditio perfecta; even more these sciences and natural philosophy have so close an affinity with one another, that unless they assist each other mutually they can in no way preserve their own dignity. For this to happen, it is necessary, first of all, that students (auditores) of physics should, at the same time, study the mathematical disciplines; this custom, up to now, has always been retained in schools of the Society. For if these sciences were taught at another time, students (auditores) of philosophy would think, and with some merit, that they are in no way necessary for physics, and so very few would want to understand them: since, nevertheless, among experts it is agreed that physics cannot rightly be understood without them, especially as regards that part which concerns the number and movement of the heavenly orbits (orbes), the multitude of intelligences, the effects of stars which depend on various conjunctions, oppositions, and other distances between them, the division of a continuous quantity into infinite [sections], the ebb and flow of the sea, winds, comets, the rainbow, the halo [of the sun and moon], and other meteorological phenomena, the proportions of motions, qualities, actions, passions, reactions, etc., concerning which mathematicians (calculatores) write much. I omit mention of the infinite examples in Aristotle, Plato, and their more illustrious commentators, which by no means can be understood without a moderate understanding of the mathematical sciences; indeed, because of ignorance of these [sciences], some professors of philosophy have very often committed many errors, and errors most grave, and (what is worse) they have even put them down in writing, some of which would not be difficult to bring forward.
For the same reason, instructors of philosophy should be skilled in the mathematical disciplines, at least moderately, lest they run onto similar bolders with great shame and loss of the reputation which the Society has in letters.
I do not mention that professors would in this way gain great influence over their students, if they [the students] understood that they [the professors] treated the places in Aristotle and other philosophers, which pertain to the mathematical disciplines, with dignity. From this it will also happen that the students will understand better the necessity of these sciences. To this end, it would be a great help if the instructors of philosophy abstained from those questions which are of little help in understanding things of nature, and very much detract from the opinion (auctoritas) about mathematical disciplines among students, such as those [questions] in which they teach that mathematical sciences are not sciences, do not have proofs (demonstrationes), abstract from being (ens) and good (bonum), etc.; for experience teaches that these [questions] are a great hindrance to students, and are not at all useful; especially because the instructors can hardly teach them (which not just once is known from the telling by others) without bringing these sciences into ridicule.
It would also be useful if in private conversations teachers were to encourage students to learn these [mathematical] sciences, impressing on them their necessity, and not, on the contrary, to lead them away from the study of these [sciences], as many have done in previous years. In this way, there will be removed every disagreement that is observed by externs to exist among Ours, when a teaching such as this is heard in schools.
Moreover, the scholastics will be greatly inspired to study these [sciences], if in every month all the philosophers gather in some one place, where one of the students will offer a brief appreciation of the mathematical disciplines, and then with one or two others will explain some problem of geometry or astronomy; this also will be pleasant to the students and of use for the humanities, and such problems are able to be found abundantly; or, let him explain some mathematical passage from Aristotle or Plato, for such passages are not isolated among them; or, even let him offer new proofs of some of the propositions of Euclid, thought out by himself; in these places, let praise be given to those who best solve the problem proposed, or who commit the fewest false syllogisms, which occur not rarely, in the invention of the new proofs. For it would happen thus, that they would become not a little eager for these studies, when they see such honor given to them, and at the same time would understand the eminence of these same studies, and they would make greater progress in these things through this exercise.
It is possible, however, to allow for this exercise perhaps the time of one morning or afternoon discussion in a month, which is not crucial for the instructors of philosophy; since this happens at most merely seven or eight times in a year, or the morning time of one day of vacation, or at least at the hour at which the mathematical disciplines are customarily taught.
Furthermore, around the end of the course of philosophy, those who wish to receive the honor (laurea) of master or doctor ought to be examined about mathematical topics, in the customary way of some other academies; at this examination, let there be present along with the other professors of philosophy, the professor of the mathematical disciplines.
In the hand of Father Brunelli. Written by the hand of Father Christopher Clavius and must be observed diligently.
It was proposed last year that, for the advancement of mathematical studies in our Society, which were already almost neglected, those who were to lecture on this science, were excused from teaching grammar for this reason, that during the first year after finishing the course of philosophy, they might study this [science] more thoroughly at home, and then teach publicly for one or two years; this [proposal] in fact was seen to be useful; it even has begun to be done in part, and its greatest usefulness will be not only for fostering this discipline, but also for adorning and augmenting it for other [disciplines]; and finally it has been demonstrated [to work], and all bear witness to this fact. But one thing seems to be troublesome about this matter. Namely, the most talented men (among whom are those who for greater service of our Lord and the good of the Society should be chosen for a ministry such as this) usually complete philosophy as adolescents, since, generally, those having the greatest age have barely attained 24 years. In addition, it seems not to be expedient for them (during the year in which they teach) that, since this ministry requires the space of one hour or two, they use the remaining [time] at their own discretion, as it seems [to happen]. Neither is it for the usefulness of the school (which is weighed down and disappointed and in which the teachers are practically boys), nor for the good name of the Society to be always using human beings such as these for explaining the more important disciplines.
For this reason, it possibly should happen that those who are chosen for this ministry [of teaching mathematics], after completing the course of philosophy, should study for an entire year at home these things which they would have taught at that time as has already been determined earlier, then right away to study theology, and then finally to teach the mathematical disciplines for the length of time which they would have taught earlier: for, in addition, men who are already of a mature age, both priests and theologians, would give honor to the chair [of mathematics] rather than degrade it, and are able to be trusted to themselves more assuredly in making use of their spare time. In addition, these advantages, not commonplace, would exist:
First. They [the students] would be able to be present at philosophical disputations, whether at the monthly [public] ones or even at private ones, and by participating [in them] to be helped themselves and stimulated, and even to help others. But if they teach before they are theologians, it would not be possible for this to occur. And, though they excel in talent, being merely students of philosophy, it would scarely be possible for them to be able to offer anything worthwhile at a disputation, and perhaps it may [even] be annoying to the teacher of philosophy to be challenged by them [the students] and to dispute with them.
Second. Regarding that time which remains after the task of teaching: By this very fact—that they have already studied theology—it be more useful if they spend [this time] by studying philosophy again and by understanding a text of Aristotle, than if perhaps they had previously devoted their attention most diligently in this same study. For, age itself offers more judgment, and the study of theology both sharpens and stimulates talent in a wonderful way. Moreover, it seems to be more proper to the Society for the glory of our Lord, that our scholastics make progress in studies to the greatest extent as can occur, especially by using the same time and effort.
Third. During this time, they [the students] can also be in charge of philosophy repetitions at home, and by this labor lighten the load of the [philosophy] teachers, who otherwise would now have even a heavier burden than before. In this way, this [proposal] will be a great asset.
Fourth. This [proposal] is better accommodated by far to studying mathematics. For in school they [the students] have already heard about the first six books [of Euclid], so they can start studying from the seventh [book] until the twelfth inclusive; then [they can] add the spherical elements of Theodosius, and some of the known [works] of Apollonius; this can easily enough be done in one year if they attend two lectures each day, which also should be done; then, also, during the four year period of theology, in the same way as the future teachers of grammar are given practice in domestic academies whenever there is nothing scheduled after the mid-day meal, so these [future mathematics teachers] also for an hour after the mid-day meal, at the same time [as the others], when the teaching has finished, should have classes in the theory of planets, gnomonics, the astrolabe, something from Archimedes and from algebra, with the material distributed as in a cycle ( orbs), and thus to come to their teaching better instructed; this will be as an ornament and useful also for the remaining studies.
It behooves that especially those be chosen who, the remaining things being equal, are outstanding in talent, diligence, and in their liking for these [mathematical] sciences, and in the way of teaching to all others, but not those who will surpass others in agreeableness (gratia). And the judgment of these matters is to be sought in earnest from those who manage them in these respects. For it sometimes happens that some, either because they are not thus inclined or because they have not been made for this study by nature, advance well enough in other subjects, but are ill-suited for mathematics.
This one thing is seen to present a difficulty to this practice: We need mathematics teachers (for whose training all this is to be done), yet we do not see from where we will be able to get those who will teach in the meantime, while those who will [eventually] teach mathematics complete their own studies. Truly this should not inhibit this [proposal]; this year there are theologians completing their studies among whom there are those who have never taught, and who willingly would study mathematics that others may teach, who are able to take their place in the meantime; and lest, while one is teaching, another uselessly wastes time, they [the students] can, in the meantime, revisit their own studies in the Ordo Studiorum and our Constitutions.
In the hand of Fr. Brunelli. Must be observed.
NOTE: Translation of Documents 34 and 35 by Dennis C. Smolarski, S.J. Translations compared with the partial translation of Document 34 found in A. C. Crombie, “Mathematics and Platonism in the Sixteenth-Century Italian University and In Jesuit Educational Policy,” PPISMATA (Prismata): Naturwissenschaftsgeschichtliche Studien (Festschrift für Willy Hartner), pp 62-94 (particularly pp 65-66) and with the translations of Documents 34 and 35 found in the Bulletin of the American Association of Jesuit Scientists (Eastern Section), v. 18, n. 4 (May 1941) (For Doc. 34, “A Method of Promoting Mathematical Studies in the Schools of the Society,” pp [203]-[206]. For Doc. 35, “On Teaching Mathematics,” pp [206]-[208]). The latter translations werepresented by Edward C. Phillips, S.J. and made by Edwin Cuffe, S.J. with the advice of Edward H. Nash, S.J.