As late medieval merged into early modern times, European men (almost always just men) started to correct serious mistakes that earlier generations had made about the natural world around them and the celestial bodies above. Some of these mistakes stemmed from religious beliefs, others from the writings of the ancient Greek philosophers and their successors. By questioning received authority, carefully examining natural phenomena, making complex mathematical calculations, and faithfully recording their findings these men transformed a scholastic philosophy into a new natural philosophy. The intellectual freedom achieved by this practice of scientific demonstration (as some of them were beginning to call it) was revolutionary in its magnitude, but not in the speed with which it took place.
Just as the transition from scholasticism and faith-based cosmology was evolutionary, so too the transition from belief in astrology, alchemy, and an ability to communicate with supernatural powers as legitimate tools for interacting with the natural world. Surprisingly, this second transition — scientists' rejection of hermetic beliefs — was even more gradual than their rejection of received truth from ancient authorities and of religious superstition (the "vain religion" of the schoolmen, as one of them put it).
The men whom we now call scientists would study the real world using careful observation and rigorous measurement, but they would also, for example, use astrology to cast one another's natal charts. And the men and women who paid the bills — aristocratic or even royal patrons, wealthy merchants, and large land owners — expected their scientist clients to produce marvels — things out of the ordinary — to show off to their friends. Or, just as likely, they expected predictions of future events that might be advantageous to them.
On the sheet of paper I've reproduced below Galileo Galilei sketched the beginnings of a birth chart for a patron, Cosimo II, which Galileo used to show that Cosimo's future was an auspicious one. At the bottom of the page he drew the moon in its waxing phase at it appeared on January 19, 1609.
There are other instances of this service of natural philosophers to those they wished to flatter. For example Johannes Kepler created horoscopes for the Holy Roman Emperor, Rudolf II, and Tycho Brahe made annual charts for the Danish king, Fredrik, and the royal princes. Here is one of the birthday charts that Tycho Brahe made for Prince Christian in 1577.
Early modern natural philosophers and mathematicians would also prepare genitures for themselves and their friends. Without at first recognizing what they were, I encountered these tables when working in manuscript collections containing correspondence among Isaac Newton and other prominent mathematicians of the late seventeenth century. I also noticed that a writer of brief biographies, like the gossipy John Aubrey, might inquire about the exact moment of a man's birth, not having reason to do so except the making of an accurate birth chart for him.
The Elizabethan magus, John Dee, presents an extreme example of a mathematician and astronomer who was also a self-proclaimed astrologer, alchemist, and practitioner of magic arts. Dee, among many other similar works, cast a horoscope to determine a favorable day for the coronation of Queen Elizabeth. Here's a chart that John Dee made for himself.
Here's a final example of belief in astrology, astronomy, and magic persisting alongside the new natural philosophy. In the mid-1650s a man named John Webster (not the famous author but a chaplain of Parliamentarian forces in the English Civil War) wrote a book criticising the archaic curriculum of the English universities. In it he said the subjects taught were hopelessly — indeed dangerously — out-of-date and devoid of usefulness. Rather than forcing students to learn dead languages or memorize the writings of Aristotle and other ancient and more recent scholastic authors, he says they should be taught mathematics and experimental science.
Webster says "Surely natural Philosophy hath a more noble, sublime, and ultimate end, than to rest in speculation, abstractive notions, mental operations, and verball disputes: for as it should lead us to know and understand the causes, properties, operations and affections of nature..." He acclaims applied mathematics — which, he points out, benefits merchants, mariners, surveyors, mechanics and others — and he condemns theoretical mathematics as merely speculative and abstract. He heaps praise on chemistry, physics, and medicine as subjects of study that are "sublime, and never sufficiently praised."
But he gives as much in the way of accolades to the esoteric subjects of magic, alchemy, and astrology. He calls magic a "noble, and almost devine Science." To him, alchemy is "the most admirable and soul-ravishing knowledge of the three great Hypostatical principles of nature." And astrology is "high, noble, excellent, and useful." As you see in this extract, he lets himself be carried away on the subject.
"Celestial Offerings: Astrological Motifs in the Dedicatory Letters of Kepler's Astronomia Nova and Galileo's Sidereus Nuncius" by H. Darrel Rutkin in Secrets of nature: astrology and alchemy in early modern Europe ed. by Anthony Grafton (MIT Press, 2001)
Galileo's Astrology by Nick KIollerstrom on skyscript.co.uk
How Galileo Dedicated the Moons of Jupiter to Cosimo II de Medici by Nick Kollerstrom, The Inspiration of Astronomical Phenomena, Proc. of 4th INSAP Conference, Oxford, Ed. N. Campion, Bristol 2004, pp. 165-181.
Tycho Brahe och Astrology
John Dee's "Mathematicall Praeface": A Sixteenth Century Classification of the Mathematical Arts and Sciences by Charles St. Clair, Norman
The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara by John Dee from Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Isaac Newton and the Transmutation of Alchemy An Alternative View of the Scientific Revolution by Philip Ashley Fanning (North Atlantic Books, 2009)
Academiarum examen, or the examination of academies, wherein is discussed and examined the matter, method and customes of academick and scholastick learning by John Webster (London, Printed for Giles Calvert, 1654)
"Education" by J.W. Adamson in The Cambridge history of English literature , in 18 Volumes (1907–21) Volume IX, From Steele and Addison to Pope and Swift ed. by Adolphus Ward and Alfred Waller (Cambridge, the University Press, 1912)
"Brief lives", chiefly of contemporaries by John Aubrey, ed. by Andrew Clark, Andrew, Volume: 1 (Oxford, Clarendon Press, 1898)
"Brief Lives" by John Aubrey, ed. by Andrew Clark (Oxford, At the Clarendon Press, 1898)
Biographies in John Aubrey's Brief Lives
Only 26 and already a professor! in Renaissance Mathematicus
"Scientific Studies in the English Universities of the Seventeenth Century" by Phyllis Allen, Journal of the History of Ideas, Vol. 10, No. 2 (Apr., 1949) Stable URL: http://www.jstor.org/stable/2707416
"John Dee" by Thompson Cooper in Dictionary of national biography ed. by Leslie Stephen and Sidney Lee (Smith, Elder, & co., 1888)
"John Dee and His Supplication to Queen Mary" by P. Evans Lewin, Woolwich Public Libraries in The Library world, Vol. 5 (Library Supply Co., 1903) Extract: 'Whilst at Cambridge he only slept four hours every night, and spent eighteen hours of the day in study. So great was his knowledge, that his acquaintance was eagerly sought by such men as Gemma Frisius, Mercator, and Gaspar a Mirca, all of whom he visited in his twentyfirst year. Even at this period he was looked on askance, for he relates that in 1547 he "sett forth" at Trinity College a Greek comedy of Aristophanes, " with the performance of the Scarabaeus, his flying up to Jupiter's palace with a man and his basket of victuals on her back, whereat was great wondering and many vain reports spread about." This, probably, was only a piece of stage mechanism suitable to the crude ideas of the time and in keeping with Greene's instructions in "Tamburlaine "—" exit Venus; or if you can conveniently let a chair come down from the top of the stage and draw her up."'
"The Mistaking of 'the Mathematicks' for Magic in Tudor and Stuart England" by J. Peter Zetterberg in The Sixteenth Century Journal, Vol. 11, No. 1 (Spring, 1980), pp. 83-97. Stable URL: http://www.jstor.org/stable/2539477
"Science and Education in the Seventeenth Century: The Webster-Ward Debate" by G. Allen; reviewed by Theodore M. Brown in Isis, Vol. 64, No. 3 (Sep., 1973), pp. 422-424. (The University of Chicago Press) Stable URL: http://www.jstor.org/stable/229755
Hermeticism in wikipedia
John Aubrey in wikipedia
Brief Lives in wikipedia
 The quote comes from Academiarum examen, or the examination of academies, wherein is discussed and examined the matter, method and customes of academick and scholastick learning by John Webster (London, Printed for Giles Calvert, 1654)
 "John Dee" by Thompson Cooper in Dictionary of national biography ed. by Leslie Stephen and Sidney Lee (Smith, Elder, & co., 1888)
 Webster's diatribe was criticized by scholars who pointed to instances where the universities were teaching the new subjects. But as Phyllis Allen and others have pointed out the chief emphasis of education at Oxford and Cambridge remained as it had been. Mathematics and natural philosophy were emphatically secondary subjects of education. "As a rule, the tutor rather than the college had the greatest influence upon the student's work. A good tutor could do much for his pupil by making wise choices in the question of reading matter. Most tutors were not interested in the new experimental sciences, and some even overlooked the work of ancient scientists. One Cambridge tutor insisted that 'Mathematics and Natural Philosophy [were] not to be hurried.' Judging by the small amount of either that the average student seems to have learned, most tutors must rarely have found time to pursue these studies in an unhurried manner. ... Aside from these classics [Aristotle, Ptolomy, Euclid, ...] there were a few modern texts available. In algebra, the more advanced undergraduate could read Thomas Hariot's Artis Analyticae Praxis (London, 1631), in which he introduced Francis Vieta's methods to England. In arithmetic, a popular text was Edmund Wingate's Arithmetique Made Easie (London, 1630). For geometry, Henry Billingsley's translation of Euclid was used, together with Christopher Clavius' commentaries upon the same author, and John Speidell's Geometrical Extraction (London, 1616), which John Aubrey says "made young men have a love to geometry." -- "Scientific Studies in the English Universities of the Seventeenth Century" by Phyllis Allen, Journal of the History of Ideas, Vol. 10, No. 2 (Apr., 1949) Stable URL: http://www.jstor.org/stable/2707416
 Webster cites John Dee as a main source for his comments on the value of applied mathematics and praises him as "that myrror of manifold learning." -- Academiarum examen, or the examination of academies, wherein is discussed and examined the matter, method and customes of academick and scholastick learning by John Webster (London, Printed for Giles Calvert, 1654)