William C. MichaelWilliam C. Michael

Mr. William C. Michael, O.P., is the founder and headmaster of the Classical Liberal Arts Academy and a Lay Dominican in the Catholic Church. Mr. Michael received an honors degree in Classics and Ancient History from Rutgers University and is a member of Phi Beta Kappa. He led the CLAA to accreditation with Cognia in 2024. He has been studying and teaching the classical liberal arts for over 30 years.

Eucild’s Elements – Book I, Common Notions or Axioms

by

Return to QRV-321 Classical Geometry I QRV-321 Lesson 003 – Book I, Common Notions I. INTRODUCTION This lesson studies the five Common Notions placed at the head of Book I of Euclid’s Elements of Geometry. Every science is built out of two kinds of statements: those that are proved, and those from which the proofs are made. The statements that are proved are called propositions, theorems and problems. The statements from which proofs are made are called principles. A science cannot prove everything, because every proof must start from something already granted; if nothing were granted, nothing could ever be … Continue

Eucild’s Elements – Book I, Postulates

by

Return to QRV-321 Classical Geometry I QRV-321 Lesson 002 – Euclid Elements, Book I, Postulates I. INTRODUCTION This lesson studies the five postulates set down at the beginning of the first book of Euclid’s Elements. Geometry is the science of continuous magnitude — of lines, angles, surfaces and solids considered according to their measure, shape and position. Like every science, geometry proves its conclusions. But nothing can be proved out of nothing. Every proof rests on something already granted. The postulates are among the things granted at the beginning of geometry, before any proposition is proved, so that everything afterward … Continue

Eucild’s Elements – Book I, Definitions

by

Return to QRV-321 Classical Geometry I QRV-321 Lesson 001 – Euclid Elements, Book I, Definitions I. INTRODUCTION This lesson begins the study of geometry, and it begins where every science must begin: with definitions. Geometry is the science of continuous magnitude — that is, of extension, of what has size and shape. Arithmetic studies multitude, which is number, and number is discrete: it is made of units that are separate from one another. Geometry studies magnitude, which is continuous: its parts hold together and are not separated by anything. Lengths, surfaces, and solids are continuous magnitudes, and geometry is the … Continue

Eucild’s Elements – Book I, Proposition 01

by

Return to QRV-321 Classical Geometry I In this lesson, we will study Euclid, Book I, Proposition 1. I. ENUNCIATION The proposition we are now to contemplate is the first of the Elements. Euclid states it in these words: On a given finite straight line to construct an equilateral triangle. Let us first understand the words themselves, before we ask why we should care to know this. A straight line, in Euclid’s usage, is not what we today call a “line” stretching endlessly. It is what we would call a segment: a length with two ends. He calls it finite to … Continue

CLAA Algebra I – The Unit of Measure

by

Return to MAT-351 Algebra I MAT-351 Lesson 004 – The Unit of Measure In the last lesson we learned that a quantity can be measured only by a quantity of its own kind, and that to measure is to find how many times one quantity is contained in another. We spoke there of the known quantity we lay beside the unknown one, but we did not name it. It has a name, and we shall learn it now. Ray gives the definition: one of the single parts of which a quantity of multitude is composed is called the unit of … Continue

CLAA Algebra I – Quantity Is Measured Only by Quantity of Its Own Kind

by

Return to MAT-351 Algebra I MAT-351 Lesson 003 – Quantity Is Measured Only by Quantity of Its Own Kind We have learned what quantity is, and we have learned that quantity is considered either as magnitude or as multitude. We come now to the first rule of measuring, and Euler states it plainly: we cannot measure or determine any quantity except by considering some other quantity of the same kind as known, and pointing out their mutual relation. Let us take his examples one at a time, for the rule is easier to see in a particular case than in … Continue

CLAA Algebra I – Magnitude and Multitude

by

Return to MAT-351 Algebra I MAT-351 Lesson 002 – Magnitude and Multitude In the last lesson we learned that quantity is anything capable of increase or decrease. That definition covers a great many different things — water and money, rope and time — and we shall now see that all of them fall into two sorts, according to the way the quantity is considered. Ray gives us the two names. Quantity is called magnitude when it is considered in an undivided form. His example is a quantity of water. When we speak of the water in a cistern, we do … Continue

CLAA Algebra I – What Quantity Is

by

Return to MAT-351 Algebra I MAT-351 Lesson 001 – What Quantity Is Before we can study algebra, we must settle what algebra is about. Every science has its own subject. The physician studies the body, the grammarian studies speech, and the mathematician studies quantity. So we begin by asking what quantity is, and we shall find that the answer, once we have it, governs everything that follows. Euler opens his Elements of Algebra with this definition: whatever is capable of increase or diminution is called magnitude, or quantity. Ray gives the same definition in his own words: quantity is anything … Continue

CLAA Homeschool Self-Evaluation Tool

by

Parents who homeschool bear a responsibility that no institution bears for them. In a school, teachers are supervised, programs are reviewed, and outside agencies conduct evaluations against published standards. In a homeschool, the parents are the administration, the faculty, and the review board at once. This is not a weakness of homeschooling. It is its nature. However, it raises a question every serious homeschooling family must eventually face: How do we know whether we are doing this well? Most parents answer that question by feel, or by comparison with public schools, with other homeschooling families, with the anxieties of the … Continue

Making Good Use of the Summer Months

by

Dear friends,  The Spring term in the Classical Liberal Arts Academy ends on Friday, June 26th — our 18th year of study. As we move towards the end of the term and the end of the school year for some, I’d like to provide some helpful information heading into the summer.  Time for Standardized Testing Parents are strongly advised to use standardized tests for two important reasons. First, standardized tests provide evidence that homeschooled children are satisfying modern educational standards while pursuing classical Catholic studies. Second, standardized tests provide helpful insights into student strengths and weaknesses, which helps with planning for … Continue