Upon those logical arguments that they set in opposition, it is to be said in general, that they, because of our finite nature, and God’s infinity and truth, are to be put in subjection to Revelation, according to 2 Corinthians 10:5. Indeed, logical arguments of this sort frequently miss the mark in the Material, since the Infinite God is reduced to the order of finite creatures; and our Perception is considered as the norm of revealed Mysteries. Thus you would incorrectly say: I myself am not able to comprehend that the eternal, spiritual, infinite, and singular God has an eternal Son by Communication of Essence, and that in the same, single Essence the Spirit is also to be acknowledged as ἐκπορευόμενον/proceeding from the Father. For you will not be able to perceive how an Essence is able to exist without a producing cause, which nevertheless is consonant with the Independent God, and without which you would have to deny all conception of Deity. The Most Illustrious VRIESIUS, in his Exercitatione Rationali XI, § 3, piously observes: “I do not at all deny that I do not clearly and distinctly conceive how such a threefold manner of subsistence consists with absolute Simplicity: but it by no means follows hence, that the former is not actually able to consist completely with the latter. I do not doubt that those things would be readily harmonized by us, if we should deeply and with sufficient vividness understand, what those Personalities are formally, and in what manner they are properly distinguished from each other and from the Essence, and how they subsist in the same singular Nature. But, if those things were thoroughly looked into, they would have nothing of Mystery left to them, etc.”: add the Most Distinguished NIEUWENTYT’S Gronden van Zekerheid, part V, chapter V, § 28, pages 455, 456, compared with chapter IV, pages 407-437. Already of old HILARY also rightly taught, de Trinitate, book I, chapter XVIII, opera, column 777: “Let us concede to God the Knowledge of Himself, and let us attend upon His words with pious reverence. For He is a sufficient witness, who is not known except through Himself.” That in matters regarding the Mystery of the Trinity it is to be believed, it is not to be asked, why? teaches ATHANASIUS, in Epistle II ad Serapionem de Spiritu Sancto, opera, tome 2, page 19: Ὡς γέγραπται, πιστευέτω τὶς, καὶ μὴ λεγέτω, διατί οὕτως καὶ μὴ οὕτως; ἵνα μὴ τοιαῦτα διαλογιζόμενος ἄρξηται μελετᾶν καὶ λέγειν· ποῦ οὖν ἐστιν ὁ Θεὸς, καὶ πῶς ἐστι; καὶ λοιπὸν ἀκούσει, εἶπεν ἄφρων ἐν καρδίᾳ αὐτοῦ, οὐκ ἔστι Θεὸς. τὰ γὰρ τῇ πίστει παραδιδόμενα, ἀπεριέργαστον ἔχει τὴν γνῶσιν. εἰ γοῦν μαθηταὶ ἀκούσαντες, βαπτίζοντες αὐτοὺς εἰς τὸ ὄνομα τοῦ Πατρὸς, καὶ τοῦ Υἱοῦ, καὶ τοῦ Ἁγίου Πνεύματος, οὐ περιειργάσαντο, διατί δεύτερον ὁ Υἱὸς, καὶ τρίτον τὸ Πνεῦμα· ἢ διατί ὅλως τριάς; ἀλλ᾽, ὡς ἤκουσαν, ἐπίστευσαν, As it is written, Let one believe, and not ask, Why thus and not thus? lest, reasoning upon such things, he begin to take thought and ask, Where is God? and how is He? and then hear, The fool hath said in his heart, There is no God: for the things delivered to faith convey an artless knowledge: when the disciples heard the words, Baptizing them in the name of the Father, and of the Son, and of the Holy Spirit, they did not impertinently ask, why the Son is the second, and the Spirit is the third, or why all together three? But, as they heard, so they believed: compare GROTIUS, de Veritate Religionis Christianæ, book III, § 12.
The same Logical Arguments often miss the mark in their Syllogistic Form also.
With the utmost brevity, our AUTHOR in his Medulla Theologiæ furnishes the best solutions to the artifaces of our Opponents, which it will have been sufficient to have comprehended in so great a Mystery. BELLARMINE endeavors to discuss the same at length, tome I of Controversiarum, book II, de Christo, chapters VII-XVIII, columns 389-408; likewise PETAVIUS, Dogmatibus theologicis, tome 2, book III, chapter IX: see also SPANHEIM, Decadum Theologicarum IV, § 6, opera, tome 3, columns 1213, 1214; ARNOLDI’S refutationem Catecheseos Racovianæ, on chapter I, de Cognitione Dei, questions 21, 22, § LXXVIII-LXXXVII, pages 112-115.
α. [That, from the pattern of Created things, Essence is multiplied Individually according to Subsistence:] see the words of Goslawski and Volkelius in HOORNBEECK’S Socinianismo confutato, tome 1, book II, chapter V, section III, page 452, 453; and Vorstius in the notis on Disputation III de Deo, page 208. But we note with our AUTHOR, that that has a place in finite things, not in the Infinite Being: compare MACCOVIUS’ Metaphysicam, book II, chapter I, pages 181-186, 191, 192, 194, 195; and the Notas of HEEREBOORD on the passage cited, as also pages 189-191. That there is rightly here an ascent unto the Infinity of the divine Essence, so that this argument’s lack of logical connection might be shown, the Most Illustrious VAN MASTRICHT also observes against Wittich, Gangræna Novitatum Cartesianarum, posterior Section, chapter XVIII, § 2-5, pages 326-328.
β. [They now conclude that, from the Essential Unity of the Persons, all Distinction between them would be Removed.] If they would prove this Syllogistically, by saying: The Father is God, and the Son is God; therefore, the Son is the Father: it is able to be given in reply, This Syllogism errs in Form, since it is in the second figure, in which one of the premises, together with the conclusion, ought to be negative.
Neither is it able to be Excepted, that this is an Expository Syllogism, and that this is also valid, although it is composed of affirmations only: for one may deny this, since God is not a predicate completely incommunicable and proper to only one Person, but common to three Persons. But if you change the Form of the Argument, and display it in the third figure in this way:
The divine Essence is the Father,
The divine Essence is the Son,
Therefore, the Son is the Father;
as if this also were an expository Syllogism: I Respond this is not actually an expository Syllogism, but a Syllogism that proves nothing, since it consists, 1. of particulars only, and, 2. has four terms: because, 1. the middle term, the divine Essence, is not altogether incommunicable, as it ought to be in Syllogisms of this sort; and, 2. it is taken in diverse senses in the major and in the minor. For in the major it is taken, either universally and completely in this sense, The divine Essence is the Father, that is, Whatever or Whoever the divine Essence is, and thus it is false; for the Son and the Holy Spirit are not the Father: or All the divine Essence is the Father; but all is is syncategorema, and may not be predicated of that which is altogether singular, and one and one only in number: or particularly, just as also in the minor, in this sense; Someone that is the divine Essence the Father is; and thus the major premise is true, but it concludes nothing, because the premises are particulars only.
At the same time, the sinew of this entire Objection is cut in a few words, when with our AUTHOR we say that, with the Numeric Unity of the divine Essence not withstanding, the divine Persons are distinguished by Characteristic Properties: compare GREGORY NYSSEN’S de eo quod non sint Tres Dii, opera, tome 3, pages 27, 28; HOORNBEECK’S Socinianismum confutatum, tome 1, book II, chapter V, section III, pages 452, 453; MACCOVIUS’ Metaphysicam, book II, chapter V, pages 20, 21, 23, 24, with HEEREBOORD’S notis.
γ. [Now, that from Personality distinguished from Essence, there would be Four Infinite Essences.] Valentinus Gentilis was reasoning so; see HOORNBEECK’S Apparatum ad Controversiam Socinianismi, page 22: so also the Socinians. I Respond that Personality is not Being, but a Mode of Subsistence, as it was seen in § 3 of this Chapter: compare MACCOVIUS’ Metaphysicam, book I, chapter IV, page 14, chapter X, pages 88, 89, book II, chapter I, page 194, with HEEREBOORD’S notis. Similar reasoning is set forth by Röellius in Judicio Ecclesiastico laudato, chapter III, § 4, pages 73, 74, to which there is a response in § 9, page 93.
δ. [Now, that from the perfect possession of Deity in each, there would be Three Individuals, etc.] The Response is, that the Individual Persons do indeed have the Essence and Essential Perfections, yet not all the Personal and Relative Properties: compare STAPFER’S Theologicæ polemicæ, tome 3, chapter XII, § 232, 233, page 486.
STAPFER in his Theologia Polemica, tome 3, chapter XII, § 227, pages 481, also reviews an Algebraic argument, advanced by the enemies of this Trinity against this Mystery, from Polykarp Leyser in his Apparatu litterario Societatis Colligentium, page 212, whereby they busy themselves to demonstrate that the Mystery of the Trinity is opposed to reason, and dashes against the Canon, that it is impossible that three perfectly equal things are able to constitute a fourth, which is not another, nor is any greater than one of these three: or a Part is not able to be equal to the whole. The words of Leyser are: “Reason objects: In the Father is the whole divine substance and essence: in the Son is the whole divine substance: in the Holy Spirit is the whole divine essence. The Father is not the Son nor the Holy Spirit, and the Son is not the Holy Spirit; Therefore, there are three divine Essences. So that I might evince that this Mystery is contrary to reason, I add a Mathematical Demonstration:
Father = a.
Son = b. Whole divine Substance = x.
Holy Spirit = c.
According to the hypothesis (of the Trinitarians):
a = x.
b = x.
c = x.
Therefore, a + b + c = x + x + x = 3x.
But according to the understanding of orthodox opinion the Father, the Son, and the Holy Spirit are one divine Essence, that is,
a + b + c = x.
Therefore, it would be x + x + x = x, or 3x = x.
But it is contrary to reason, that a part is equal to the whole.
Therefore, the Mystery of the Trinity is also against reason. Thus it has been demonstrated.
But on this argumentation it comes to be observed, 1. that Algebraic Calculations are only able to be applied to Quantities; but, since to the Trinity are applicable neither parts nor material quantity, which would have some proportion of magnitude or other Attribute of Body, neither is this calculation able to be applied to this Mystery: for that has nothing equal or similar in physical things. 2. The Orthodox do not acknowledge this Proposition, that the Whole divine Substance and Essence is in the Father, for example, if by the Whole divine Substance and Essence you understand the Essence with all its modifications: for the modification of Essence or the characteristic Property modifying the Essence is one thing in the Father, another in the Son, and another in the Holy Spirit. But, although the Father, the Son, and the Holy Spirit are hence one and another person, they are not one and another thing: but one and the same divine and spiritual Essence enjoys three distinct modifications, which characterize the Essence in the individual Persons, and are incommunicable to the other Persons. Which matter, having nothing similar in created things, is nevertheless clearly revealed in Sacred Scripture, but at the same time is set forth to us as a Mystery; which it is fitting piously to believe, rather than anxiously to scrutinize: neither is there to be a detraction from the veneration owed to infinite spiritual Essence by a comparison of the same with things material and enjoying quantitative dimension. 3. At the same time, if it be allowed to make use of Algebraic calculation in investigating this Mystery, the orthodox opinion, set forth according to the understanding of the Reformed Church and of others that duly hold the doctrine of the Trinity, could also be defended by Algebraic demonstration against the Objection set forth above, as has been done by the Most Illustrious CARPOV in his Revelato Trinitatis Mysterio method demonstrative proposito, in STAPFER’S Theologiæ Polemicæ, tome 3, chapter XII, § 229, pages 483.
That is, according to the understanding of the Orthodox the whole divine Substance with all its modifications or personal characteristics, indicated by x, is equal to the Father, the Son, and the Holy Spirit taken at the same time: or x = a + b +c.
With this posited, if any Person be taken twice, it is the same as if the whole Essence were taken twice, and the whole Essence taken twice will be equal to the three Persons taken twice, or:
Therefore, 2x = 2a + 2b +2c.
Now, if equal things be subtracted from equal things, the remainders will be equal.
Hence, if the sum of the letters or quantities a, b, and c, are equal to quantity x, and if from one side or the other is subtracted the sum of quantities b and c, quantity a will remain equal to quantity x, with the sum of quantities b and c removed; and in the same manner also the matter will hold with b, with the sum of a and c removed, and also with c, with the sum of a and b removed. In this manner:
God the Father (a) will be equal to x, or the divine Essence, with the Son (b) and also the Holy Spirit (c) subtracted, or:
a = x – b + c.
Likewise, b = x – a + c.
And, c = x – a + b.
Again, if we should add equal things to equal things, the sums thus arising are equal; and so the sum resulting from a, b, and c will be equal to triple x, with the sum of twice a, twice b, and twice c, removed, or:
Therefore, a + b + c = 3x – (2a + 2b + 2c).
Now, equal are those things that are able to be substituted one for another with the quantity preserved: whence also into the place of the sum of twice a, twice b, and twice c, is able to be substituted twice x, since it was seen above that 2x = 2a + 2b + 2 c.
Therefore, 3x – (2a + 2b + 2c) = 3x – 2x.
Whence, finally, remains what was above in the Proposition of the orthodox opinion,
Therefore, a + b + c = x.
Or the three divine Persons taken together constitute the numerically one divine Essence or Substance.
Thus, by the grace of God it is not necessary to concede that the Contradictions, which the Anti-Trinitarians rashly contrive from the doctrine of the Trinity, are hardly able to be resolved, as our AUTHOR observes others imprudently to do: compare MARESIUS’ Systematem Theologicum, locus III, § 41, note d, page 124; and his Dissertationem de Abusu Philosophiæ Cartesianæ, § 113, number 23, page 88; WITTICH’S Theologiam Pacificam, § 6, pages 5, 6; VAN MASTRICHT’S Gangrænam Novitatum Cartesianarum, posterior Section, chapter XVIII, § 12-15, pages 335-338; LEYDEKKER’S Facem Veritatis, locus IV, controversies II, pages 201-204. On § 19 and 20 see also VAN CATTENBURGH, Spicilegio Theologiæ Christianæ, book II, chapter XVII, section IV, page 203-206. On § 20 also thoroughly compare LILIENTHAL, Oordeelk Bybelverklar, chapter IV, § 20-27, 38-50, part 1, pages 266-275, 291-314.
 Bernard Nieuwentyt (1654-1718) was a Dutch Reformed theologian and Cartesian philosopher.  In the second figure, the middle term is the predicate in both premises.  A Syllogism is Expository or Singular, when its middle term is singular.  That is, language signifying quantity.  Polykarp Leyser IV (1690-1728) was a German Protestant scholar, studying theology and history, and taking degrees in philosophy, medicine, and law.  Jakob Carpov (1699-1768) was a German Lutheran philosopher, theologian, and educator. He is remembered for his application of Wolffian philosophy and mathematical demonstration to theology.