Phelipe Medrano
Phelipe Medrano was an 18th-century Spanish nobleman and mathematician of the House of Medrano. A knight of the Order of Santiago, he was active at the Bourbon court during the reign of Philip V and is known for his work in arithmetic and combinatorial mathematics, produced within the institutional and intellectual culture of early Enlightenment Spain. He is best known as the author of Quadrados mágicos, que sobre los que figuraban los Egyptcios, y Pythagoricos, a printed mathematical treatise dedicated to Queen Isabel Farnese. The work presents more than one hundred magic squares constructed through systematic numerical methods and was formally approved by Diego de Torres Villarroel, with prefatory poetic contributions from members of Madrid literary academies. A subsequent unpublished manuscript, Solución general y natural de los Quadrados mágicos, expands his mathematical method into a structured sequence of definitions and demonstrations, situating Medrano among the learned noble authors contributing to mathematical discourse in Bourbon Spain.
Life and Background
Phelipe Medrano was the son of Pedro Medrano, a knight of Santiago, an illustrious member of the House of Medrano, and crown official. The House of Medrano were deeply embedded in the Order of Santiago, with multi-generational nobles maintaining important roles in the Order for centuries. Active in the court of early 18th-century Bourbon Spain, Phelipe identified himself as a knight of the Order of Santiago, like his father before him. While no birth record has been located, his position and status are evident in the courtly dedication, poetic endorsements, and structural design of his printed treatise, Quadrados mágicos.The House of Medrano's deep involvement in the Military Orders of Spain, particularly the Order of Santiago, culminated in 1605 when Philip III of Spain reformed the Order of Santiago with the works of García de Medrano y Castejón, enacting one of the most comprehensive reforms of their statute laws. By the 18th century, Phelipe, Pedro, and other noblemen of the Medrano family continued this tradition, holding the rank of knight of the Order of Santiago and serving on the Royal Council of Orders. Pedro was active during the same period as Giovanni Antonio Medrano, also known as Juan Fernández de Medrano, who served as royal architect and tutor to the Bourbon princes.
According to the Heraldrys Institute of Rome, the Medrano family is well known "for its antiquity, its splendor, for their military prowess and virtue and for every other value of chivalry that prospered with this family, in great numbers, magnificent and generous."
Pedro Medrano
The clearest testimony to Phelipe Medrano's lineage appears in a printed elogio included in the prefatory section of Quadrados mágicos, authored by the This praise comes from Ignacio de Loyola y Oyanguren, a celebrated writer born 9 February 1686, who inherited the title of 2nd Marqués de la Olmeda,Himself a Knight of Santiago and royal official, the marqués offers praise for the author's father, Pedro Medrano, citing his titles and offices in full:
Pedro Medrano; knight of the same Order, secretary to His Majesty, and senior second official of the Secretariat of State for the Negotiation of Italy, and of the Universal Dispatch Office with authority over the exercise of decrees.
The 2nd Marqués de la Olmeda, Cavalier of the Order of Santiago, Commander of Villa-Rubia de Ocaña, and Procurador General of said Order, personally confirms that Phelipe's work was received as a continuation of a family tradition of noble service.
In the elogio of Phelipe Medrano's Quadrados Magicos, the Marqués de la Olmeda describes Philip's father Pedro as a "living archive of history... a Caliodorus of the world," a celebrated state counselor, and the source of the author's intellectual inheritance. The elogio elaborates on this legacy with the following verses:
Since your father, in brief,
Granted you the improvement of third and fifth,
Your father, whose glory
Was a living Archive of History,
And in affairs of State,
A Caliodorus of the world, celebrated far and wide,
What wonder, then, if my rough wit
Should stand in awe of a branch sprung from such a trunk?
His father, Pedro Medrano, held the post of Secretary to His Majesty, and the highest-ranking Second Official of the Secretary of State for Italy during the reign of Charles II, having previously served as Secretary of War for the Navy.
Phelipe's father as Secretary of the Council of State
Contemporary documentation confirms his father's high-level role in Spanish foreign policy. In March 1672, Antonio de Mendoza, newly appointed envoy to Genoa, addressed Medrano's father Pedro Medrano with deference, who at that time served as Secretary of the Council of State:
I acknowledge having received notice from the President, following whose dispatch I have requested, by way of my signature, as a means of ensuring that this matter may proceed as is fitting, safeguarding my position. May God keep Your Grace for many years.
A month later, Lope de los Ríos, President of the Council of the Treasury, formally instructed Pedro Medrano to authorize Mendoza's travel allowance by royal order. This document, held in the Spanish Archives, records Pedro's responsibility for coordinating and executing diplomatic policy at the state level.
Continuity
While Pedro's distinguished service took place under the last Habsburg monarch, Charles II, Phelipe Medrano's Quadrados mágicos appeared at the Bourbon court under Philip V, maintaining the continuity of Medrano family service across dynastic transitions. The numerical structures in Phelipe's work reflect the same values of order, balance, and hierarchy that underpinned his father's political and diplomatic roles.Works
''Quadrados mágicos'' (1744)
Phelipe Medrano, knight of the Order of Santiago and son of Pedro Medrano, Secretary of the Council of State for Italy, is best known for his 1744 treatise Quadrados mágicos, que sobre los que figuraban los Egyptcios, y Pythagoricos. Printed in Madrid and dedicated to Queen Elisabeth Farnese, the work presents over one hundred magic squares, from 3×3 to 32×32, not as mathematical curiosities but as diagrams of political harmony, spiritual hierarchy, and noble service within the Doctrine of Medrar.Endorsed by Diego de Torres Villarroel and introduced with poetic tributes from members of the Academia Poética Matritense, the book encodes a theory of governance through number. Medrano invokes Egyptian and Pythagorean traditions to frame each square as a symbolic structure of delegated rule, aligning mathematics with dynastic continuity and Bourbon pedagogy. Phelipe's mathematical treatise expanded the Doctrine of Medrar, giving new form to delegated rule, structured rise, and the visual grammar of power throughout the centuries.
Dedication to Queen Elisabeth Farnese
The dedication that opens Quadrados mágicos is a pure act of political positioning. Medrano offers his mathematical work to Queen Elisabeth Farnese not as a book, but as a "sacrifice" grounded in sacred geometry and planetary tradition. He invokes the ancients, "those once dedicated... to the Planets," only to reject their "superstitious error," insisting that his offering is rational, Christian, and loyal. The structure of the dedication, like the squares themselves, is ordered, ascending, and centered on a single sovereign axis: the Queen.The work's dignity, he writes, lies not in its content, but in the royal hand that receives it:
At Your Majesty's feet I imprint the lips of my reverence and raise this small gift I offer. I present these Magic Squares, once dedicated by Egyptians and Pythagoreans to the Planets, theirs founded on superstition and error, mine securing dignity in its object and dedication. They believed such Squares granted protection, an erroneous belief, for the credulous were left exposed to danger. But I, in this sacrifice, extinguish that false superstition and establish a true and constant benefit to the realm, whose superior Planet wards off danger and enriches the glory of Spanish Lords.
Medrano casts Elisabeth as "the superior PLANET," whose aspect ensures protection and noble flourishing. His language becomes a sacred geometry, elevating service into a visible diagram of advancement. In aligning classical form with Bourbon authority, Medrano transforms the magic square into a ceremonial instrument of fidelity.
''Solución general y natural de los Quadrados mágicos'' (1745)
In 1745, Phelipe Medrano authored a manuscript entitled General and Natural Solution to One of the Most Celebrated and Most Difficult Problems of Arithmetic Named Magic Squares, Founded on the Properties That Any Progression Placed in a Square Figure Has, with a Demonstration of the Operations. Significantly longer than his earlier work, it spans over 200 pages of text and tables and presents a distinctly mathematical focus. This complexity may partly account for why, to the best of current knowledge, the manuscript was never published.Poetic tributes to Phelipe Medrano
Philip was introduced in his Quadrados mágicos with poetic tributes by members of Madrid's literary elite, many of whom were high-ranking nobles, poets, and officials. These texts collectively elevate Phelipe Medrano as a paragon of mathematical virtue, enlightenment learning, and noble service. Collectively, the poetic tributes frame Quadrados mágicos as more than a mathematical treatise. Through imagery drawn from classical philosophy, biblical allegory, and the history of science, Medrano is depicted as a figure who unites erudition, noble lineage, and loyal service to the Crown. The verses consistently contrast ancient superstition with Christian rationality, recasting the magic square from an emblem of idolatry into a symbol of order, governance, and enlightened rule. In this literary context, Medrano’s work is celebrated not only for its technical precision but also as an expression of the Medrano family's enduring Doctrine of Medrar, understood as the advancement of knowledge and the state through disciplined structure and loyal service.Joseph Cañizares
Among the poetic tributes prefacing the work is a laudatory sonnet by Joseph Cañizares, Theniente de Caballos Corazas, composed in honor of the author and his mathematical achievement:
Socrates immortal, Plato divine; If they ploughed numeric seas, They had a course, for the North they found, Following the Pythagorean path. If the Tarentine of the soul's affections, dealt in numbers, By Nicomachus they found the method, In learned and wandering Arithmos.
Only you, without north or guide, By influences of Sovereign Numen, Brilliant mists, showing the day; The cheerful, vain Egyptian may say, My weak Arithmetical harmony, If it began to medrar, it is because of Medrano.
The connection between medrar and Medrano is etymological, conceptual and programmatic. Classical sources from Antonio de Nebrija to Joan Corominas confirm that the surname Medrano derives from the verb medrar, meaning "to improve," "to advance," or "to prosper." The 18th-century poet José de Cañizares made this semantic unity explicit in his tribute to Phelipe Medrano, writing: "If my weak Arithmetic shows any medrar, it is because of Medrano."
His magic squares, structured, ascending, and centered, are symmetrical manifestations of social and cosmic order. As Àngel Campos-Perales has shown, the Spanish monarchy under the Habsburgs and Bourbons relied on Medrar to curate visual grammars, including chapels, funerary architecture, heraldry, and ceremonial emblems, to create a symbolic language that legitimized delegated power and kingship within the visual field of court society. Quadrados mágicos fulfills this same function in mathematical form. Through calibrated order and recursive harmony, Medrano’s squares become a visual geometry of Medrar: noble advancement through ordered legitimacy made intelligible through number. They inscribe, in mathematical structure, what the name Medrano etymologically and doctrinally signifies: improvement, legitimacy, and structured ascent. In this way, each magic square becomes not only a solution but a visible step in the Medrano Doctrine of Medrar.
Ignacio de Loyola y Oyanguren, 2nd ''Marqués de la Olmeda''
The Elogio by Ignacio de Loyola y Oyanguren, 2nd Marqués de la Olmeda and dedicated to Phelipe Medrano in the Quadrados mágicos declares:
The Egyptians invented magic squares,
And placed their deities in them,
As though divinity resided
In whatever had quantity,
For the Supreme Deity
Knows no bounds, nor permits summation,
Their delirium ever shifting,
A new literary labyrinth.
But today your noble name
Amazes the entire world,
Lifting from such blind confusion
The undeniable being of truths.
I have gladly examined
The work you’ve shared with me;
In its progressions,
You’ve refined the combinations
So the world may see,
If one studies with care,
How distance conquered
Turns ignorance into radiance.
Your tested problems
Leave my concerns so fully satisfied,
That any studious mind
To which they are applied, wise and attentive,
Will find you have reached such depth of understanding,
That, once acquired, your knowledge becomes inherited.
Since your father, in brief,
Granted you the improvement of third and fifth,
Your father, whose glory
Was a living Archive of History,
And in affairs of State,
A Caliodorus of the world, celebrated far and wide,
What wonder, then, if my rough wit
Should stand in awe of a branch sprung from such a trunk?
Our friendship, in short, is not of recent days,
For it holds its proper place before the lady.
I’ve always known
Your playful genius in study,
And in your early green seasons,
Like a rational bee you gathered blossoms
So that those flowers would yield fruit.
Mathematics is what you descend from,
A sea of faculties, which, when grasped,
Even in narrow chambers,
Reduce to number and to measure.
There is nothing in this life
That cannot be reduced to number,
Even human fate
Finds its final unity in death.
That beautiful planet,
Which follows its luminous course,
In the zenith of its rays,
Turns its light into a foam-topped grave.
The rose, which in various views
Is seen as the living water of attentions,
When its bud is threatened,
Finds that from one unit alone it grows,
And breaking the bud’s scarlet shell,
Expands into countless forms.
The poor little spring,
Brief in its course, yet marvelous,
When it becomes a deep arroyo,
Offers a crystal torrent,
And in time, as it animates its being,
Its liquid flow multiplies.
Thus it is seen in light, in crimson, and in foam,
The multiplication, the line, and the smoke.
Books of the clearest minds
Often carry more contradiction,
But your book, in superior fortune,
Suffers neither flaw nor failing,
For to criticize it,
One must first comprehend it.
So much consequence
You place in demonstration,
That what you prove
Silences the Babel of dispute,
For in mathematics,
Only "ergo" makes the proof.
You are today a new Ganymede,
Grasping Archimedes' spheres,
Drawing from their motions
New problems from Euclid’s torrent.
And as Isidore once said
With truth on his lips and a golden pen,
That to join the whole with its parts
Is the science of sciences and art of the arts,
Let my crude tongue cease its applause,
For here, silence itself is eloquence.
The tribute to Phelipe Medrano, composed by Ignacio de Loyola y Oyanguren, 2nd Marqués de la Olmeda is a doctrinal witness, a compressed theology of number, authority, and lineage that situates Medrano's work within both the Christian tradition and the universal Doctrine of Medrar. Loyola begins by recalling that the Egyptians "invented magic squares" and placed their deities within them, as though divinity could be reduced to quantity. Against this, he asserts that the "Supreme Deity knows no bounds, nor permits summation." The elogio echoes the Medrano doctrine: number is not itself divine, but a sign of divine order. Medrano is praised as the one who purifies mathematics from superstition and re-anchors it in theology and law.
The tribute contrasts the "delirium" and "labyrinth" of pagan systems with the clarity that comes "today" through the Medrano name. To say "your noble name amazes the entire world" is not presented as flattery alone. It positions Medrano identity, rooted in medrar meaning to rise and to advance, as a public emblem that transforms confusion into truth. The name itself is presented as a doctrinal sign of order.
In describing Medrano's "progressions" and "combinations," Loyola moves from technical vocabulary to moral theology. Mathematical method disciplines not only the intellect but also desire, converting ignorance into radiance. The problems Medrano sets are "tested," and the "inheritance" of his knowledge is emphasized. His mathematics is portrayed as transmissible, forming future generations. Nobility is presented here as pedagogical as much as genealogical. The image of the "rational bee" gathering blossoms underscores this point, presenting Medrano's learning as ordered play that yields lasting fruit.
Specifically, In the Elogio, the 2nd Marquess of La Olmeda described Phelipe Medrano as a "rational bee," gathering from varied flowers to form a disciplined and ordered sweetness. The metaphor of the bee had already appeared in the Medrano tradition: in his República Mista, Tomás Fernández de Medrano compared political unity to the natural order of bees, "in whom one queen rules; as in the flock one shepherd; in the world one God; and in the kingdom one prince."
Read together, these images illustrate how Spanish society employed the bee as a recurring symbol, at once of intellectual formation in the Elogio and of political and theological order in Tomás's treatise. By drawing on the same image, Ignacio de Loyola's tribute connected Phelipe Medrano's mathematical treatise and his surname to the legitimacy and continuity of the established doctrines for Spain that the Medrano family had been cultivating for centuries.
Loyola further insists that Phelipe Medrano's book suffers "neither flaw nor failing" and that his demonstrations silence the "Babel of dispute." Proof rather than patronage confers legitimacy. In the same way that governance depends on demonstrable justice, mathematics depends on the necessity of demonstration. Medrano is described as a "new Ganymede" grasping Archimedes' spheres, a metaphor for noble ascent joined to service of sovereign authority. Loyola concludes by invoking Isidore of Seville's maxim that wisdom joins the whole with its parts, suggesting that Medrano's magic squares achieve this union by binding throne and subject, unit and sum, universal and particular.
The elogio presents Phelipe Medrano as reformer of number and demonstrator of the Doctrine of Medrar. It frames his name as a sign of order revealed to the world, lifting mathematics from superstition into truth. The reaction and amazement of the world that Loyola describes is recognition of Medrano and a precept inscribed in creation: that true nobility lies in transmitting order from the One to the many, and from the many back to the One.
Joaquín de Aguirre
His friend Joaquín de Aguirre wrote the following "on the occasion of publishing the Treatise" of the Magic Squares:
Darkness teaches, which exhales vice
In clumsy fumes that nourish error,
All that is obscure is driven away by Light,
And sacrilege transforms into Sacrifice.
An irrational reef, where the Egyptian
Treads misguided steps in bloodied ways,
Now becomes a heroic blazon, no longer a shame,
If frenzy slumbers, let judgment rise.
That which was once impossible is now conquered,
That which was mute now speaks and is explained:
That error, converted into truth.
Sacrilege is now a consecrated emblem,
It spreads nobility to the public,
The prodigious name of a Crusader.
Aguirre wrote a second sonnet dedicated to Medrano:
If faith could secure the noble effort
Of that other Philosopher who once claimed
That one soul passed through many bodies,
Then yours, Medrano, stands above his.
Madness it seemed, or error, or a foolish dream,
Yet the world itself praises you,
Though some denied it with lesser assent,
Their judgment diminished by their own pride.
Indeed, Pythagoras said the soul
Could not remain in a single form,
But the soul that animated you wins the crown.
So let them discuss, invent, write, teach, dispute,
The greatest minds now stand aside,
Asking: was that soul his, or is yours the better?
Miguel de Villa-Fuerte
Among the poetic tributes prefacing the work is a sonnet by Don Miguel de Villa-Fuerte, lawyer of the Royal Councils, composed in praise of the author and his learned achievement:
By what rule, instruction, or teaching
Humbly I ask, have you uncovered
That uncultivated path, long smothered,
Which thought alone deemed unapproaching?
If in nine Problems you staked your aim
To discern and resolve where lofty heights dwell,
The subtle art you studied well,
Revealing clear your Arguments’ flame.
Ah, do not answer how to understand you;
Jealousy more than ignorance would know you,
Let the Egyptians themselves bow down in awe.
Nearly impossible to imitate you,
None may presume to grasp what you do,
In the difficult, you are the law.
Francisco de Quadros
Don Francisco de Quadros, Knight of the Order of Santiago, offers a poetic tribute titled "Heroic Romance" applauding the tireless study of Phelipe Medrano and his accomplishment in composing the Magic Squares once devised by the Egyptians:
If the Egyptians, with errant trace,
Worshipped fabled deities' delight,
And in the unfathomable gulf of guile,
Sought to fix their foot in glory’s place,
Tearing bronze in solemn rite,
Their fraud hid under hardness bright,
Numbers they privileged with divine right,
Though no Numen they adored with grace.
Superstitious simulacra made
Votive smoke serve as torchlight’s aid,
Letting their madness burn once more.
And in their study, void of guide,
They left no trace or text behind,
So none might follow, none explore.
But what applause now may be born,
Lesser than your scorn,
If daring measured the impossible's form?
Your golden pen has sealed their mouths forlorn.
Magic, they called your noble aims,
Yet only fools would make such claims,
For in their error, none believed
A man such glory could achieve.
Arithmetic, in you its purest height,
Crowns you with its diadem bright,
And from the secret path of numbers’ might,
You pass still further when it takes flight.
The strange Problems your labor posed
Shamed curiosity’s boastful prose,
For insight prefigured all you chose,
Before the page your logic froze.