Eucild’s Elements – Book I, Definitions

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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

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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

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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

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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

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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

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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

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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

Trading the Catholic Intellectual Inheritance For Money

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When the modern State’s pedagogical reformers approach Catholics and offer them a better vineyard or a sum of money in exchange for the use of our field – not to continue producing long tested, deeply rooted Catholic educational outcomes but for the State’s own immediate needs – what is our response? 

Whether Public Charter Schools Are Fit for Catholic Communities Seeking Classical Education

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Objection 1. It seems that public charter schools are a good and prudent choice for Catholic communities in 2026. For in this age of high taxation and burdensome private tuitions, such schools grant families from a range of economic backgrounds access to rigorous, smaller-scale learning environments that often emphasize classical methods and virtue formation. Recent expansions in school choice and legal developments have made it possible for parents to direct public resources toward options that escape the worst failures of traditional school districts, thereby fulfilling the natural duty of parents to provide fitting education without impoverishing the household. Objection 2. … Continue

Making Good Use of the Summer Months

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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